Self-gratification by a²+kb² forms "with ~10% FALSE STATEMENTS" (by author) - Page 11

Raman · 2013-03-21, 19:26

The following identities can be used to study for how the different polynomials of discriminant = -4k does
combine together as products by using their own factors inside the following number field in Z[√-k] of as follows as.

Quote:

Originally Posted by Raman

$a^2+kb^2 = (a+b\sqrt{-k})(a-b\sqrt{-k})$
$ma^2+nb^2 = (a\sqrt{-m}+b\sqrt{-n})(a\sqrt{-m}-b\sqrt{-n})$
$Xa^2+Yab+Zb^2 = X\left(a+\frac{b}{X}\left(\frac{Y}{2}+sqrt{-\left(XZ-\frac{Y^2}{4}\right)}\right)\right)\left(a+\frac{b}{X}\left(\frac{Y}{2}-sqrt{-\left(XZ-\frac{Y^2}{4}\right)}\right)\right)$

While although the following identity $ma^2+nb^2 = (a\sqrt{-m}+b\sqrt{-n})(a\sqrt{-m}-b\sqrt{-n}$
is being correct , enough altogether with ,
whereas the following identity $ma^2+nb^2 = m\left(a+\frac{b}{m}\left(sqrt{-\left(mn\right)}\right)\right)\left(a+\frac{b}{m}\left(-sqrt{-\left(mn\right)}\right)\right)$
may be preferred with the following term $\sqrt{-mn}$ being available
whenever taking products of polynomial factors from different quadratic forms,
for the study of how they combine together as the product of their factors as follows as.

Two numbers 14 and 18, none of them 14 or 18 are being writable as a²+3b² form,
but their own combined product 252 is being writable as a²+3b² form,
as follows as,
252 = 12² + 3 × 6².

Two numbers 12 and 15, none of them 12 or 15 are being writable as a²+b² form,
but their own combined product 180 is being writable as a²+b² form,
as follows as,
180 = 12² + 6².

Two numbers 10 and 15, none of them 10 or 15 are being writable as a²+2b² form,
but their own combined product 150 is being writable as a²+2b² form,
as follows as,
150 = 10² + 2 × 5².

Two numbers 14 and 21, none of them 14 or 21 are being writable as a²+2b² form,
but their own combined product 294 is being writable as a²+2b² form,
as follows as,
294 = 14² + 2 × 7².

3 is not being of the form as a²+11b², but that 3³ = 27 is which is being of the form as 4² + 11 × 1².
5 is not being of the form as a²+11b², but that 5³ = 125 is which is being of the form as 9² + 11 × 2².
23 is not being of the form as a²+11b², but that 23³ = 12167 is which is being of the form as 54² + 11 × 29².

It seems likely that this following scenario cannot happen for the values of k = convenient numbers, cases
as such as cases as follows as

If k is being a convenient number, then certainly that always that if N is not of the form as a²+kb²,
then so for the following numbers N³, N⁵, N⁷, N⁹, ..., N^2x+1

as since as cases as well as

For all the values of numbers k cases , it easily follows that this thing and then stuff matters does never violates that way that following that condition that
that certainly that always that holds out that off r r r r r r r r r r
that way
if N is of the form as a²+kb²,
then so for the following numbers N², N³, N⁴, N⁵, N⁶, N⁷, N⁸, N⁹, ..., N^x, both,
ofr r r r r r r r r r r the
for r r r r r r r r r r the
N^2x+1,
N^2x

Raman · 2013-03-21, 19:28

k = 47

Primes p of the form a² + 47b²: Some primes congruent to {1, 3, 7, 9, 17, 21, 25, 27, 37, 47, 49, 51, 53, 55, 59, 61, 63, 65, 71, 75, 79, 81, 83, 89} mod 94, neither of the form 3a² + 2ab + 16b² nor 7a² + 6ab + 8b²
if N is a non-negative integer that can be written as a²+47b², then p × N can be written as a²+47b²
if N is a non-negative integer that cannot be written as a²+46b², then p × N cannot be written as a²+47b²
Primes of the form 3a² + 2ab + 16b²: Some more primes congruent to {1, 3, 7, 9, 17, 21, 25, 27, 37, 49, 51, 53, 55, 59, 61, 63, 65, 71, 75, 79, 81, 83, 89} mod 94, neither of the form a²+47b² nor 7a² + 6ab + 8b²
Primes of the form 7a² + 6ab + 8b²: Other remaining primes congruent to {1, 3, 7, 9, 17, 21, 25, 27, 37, 49, 51, 53, 55, 59, 61, 63, 65, 71, 75, 79, 81, 83, 89} mod 94, neither of the form a²+47b² nor 3a² + 2ab + 16b²

N can be written as a² + 47b² if and only if
- N is not congruent to 2 (mod 4).
- N has no prime factors congruent to {5, 11, 13, 15, 19, 23, 29, 31, 33, 35, 39, 41, 43, 45, 57, 67, 69, 73, 77, 85, 87, 91, 93} mod 94 to an odd power.
- If N is odd or N ≡ 4 (mod 8), then the (sum of exponents of prime factors of N of the form 3a² + 2ab + 16b², sum of exponents of prime factors of N of the form 7a² + 6ab + 8b²) is not (0,1) or (1,0) or (1,1) or (3,0) or (0,3).
- If N ≡ 8 (mod 16), then the (sum of exponents of prime factors of N of the form 3a² + 2ab + 16b², sum of exponents of prime factors of N of the form 7a² + 6ab + 8b²) is not (0,0) or (1,0) or (0,2).
- If N ≡ 16 (mod 32), then the (sum of exponents of prime factors of N of the form 3a² + 2ab + 16b², sum of exponents of prime factors of N of the form 7a² + 6ab + 8b²) is not (0,1).
- If N ≡ 32 (mod 64), then the (sum of exponents of prime factors of N of the form 3a² + 2ab + 16b², sum of exponents of prime factors of N of the form 7a² + 6ab + 8b²) is not (0,0).

k = 79

Primes p of the form a² + 79b²: Some primes congruent to {1, 5, 9, 11, 13, 19, 21, 23, 25, 31, 45, 49, 51, 55, 65, 67, 73, 79, 81, 83, 87, 89, 95, 97, 99, 101, 105, 111, 115, 117, 119, 121, 123, 125, 129, 131, 141, 143, 151, 155} mod 158, neither of the form 5a² + 2ab + 16b² nor 8a² + 6ab + 11b²
if N is a non-negative integer that can be written as a²+79b², then p × N can be written as a²+79b²
if N is a non-negative integer that cannot be written as a²+79b², then p × N cannot be written as a²+79b²
Primes of the form 5a² + 2ab + 16b²: Some more primes congruent to {1, 5, 9, 11, 13, 19, 21, 23, 25, 31, 45, 49, 51, 55, 65, 67, 73, 81, 83, 87, 89, 95, 97, 99, 101, 105, 111, 115, 117, 119, 121, 123, 125, 129, 131, 141, 143, 151, 155} mod 158, neither of the form a²+79b² nor 8a² + 6ab + 11b²
Primes of the form 8a² + 6ab + 11b²: Other remaining primes congruent to {1, 5, 9, 11, 13, 19, 21, 23, 25, 31, 45, 49, 51, 55, 65, 67, 73, 81, 83, 87, 89, 95, 97, 99, 101, 105, 111, 115, 117, 119, 121, 123, 125, 129, 131, 141, 143, 151, 155} mod 158, neither of the form a²+79b² nor 5a² + 2ab + 16b²

N can be written as a² + 79b² if and only if
- N is not congruent to 2 (mod 4).
- N has no prime factors congruent to {3, 7, 15, 17, 27, 29, 33, 35, 37, 39, 41, 43, 47, 53, 57, 59, 61, 63, 69, 71, 75, 77, 85, 91, 93, 103, 107, 109, 113, 127, 133, 135, 137, 139, 145, 147, 149, 153, 157} mod 158 to an odd power.
- If N is odd or N ≡ 4 (mod 8), then the (sum of exponents of prime factors of N of the form 5a² + 2ab + 16b², sum of exponents of prime factors of N of the form 8a² + 6ab + 11b²) is not (0,1) or (1,0) or (1,1) or (3,0) or (0,3).
- If N ≡ 8 (mod 16), then the (sum of exponents of prime factors of N of the form 5a² + 2ab + 16b², sum of exponents of prime factors of N of the form 8a² + 6ab + 11b²) is not (0,0) or (1,0) or (0,2).
- If N ≡ 16 (mod 32), then the (sum of exponents of prime factors of N of the form 5a² + 2ab + 16b², sum of exponents of prime factors of N of the form 8a² + 6ab + 11b²) is not (0,1).
- If N ≡ 32 (mod 64), then the (sum of exponents of prime factors of N of the form 5a² + 2ab + 16b², sum of exponents of prime factors of N of the form 8a² + 6ab + 11b²) is not (0,0).

k = 103

Primes p of the form a² + 103b²: Some primes congruent to {1, 7, 9, 13, 15, 17, 19, 23, 25, 29, 33, 41, 49, 55, 59, 61, 63, 79, 81, 83, 91, 93, 97, 103, 105, 107, 111, 117, 119, 121, 129, 131, 133, 135, 137, 139, 141, 149, 153, 155, 159, 161, 163, 167, 169, 171, 175, 179, 185, 195, 201, 203} mod 206, neither of the form 7a² + 6ab + 16b² nor 8a² + 2ab + 13b²
if N is a non-negative integer that can be written as a²+103b², then p × N can be written as a²+103b²
if N is a non-negative integer that cannot be written as a²+103b², then p × N cannot be written as a²+103b²
Primes of the form 7a² + 6ab + 16b²: Some more primes congruent to {1, 7, 9, 13, 15, 17, 19, 23, 25, 29, 33, 41, 49, 55, 59, 61, 63, 79, 81, 83, 91, 93, 97, 105, 107, 111, 117, 119, 121, 129, 131, 133, 135, 137, 139, 141, 149, 153, 155, 159, 161, 163, 167, 169, 171, 175, 179, 185, 195, 201, 203} mod 206, neither of the form a²+103b² nor 8a² + 2ab + 13b²
Primes of the form 8a² + 2ab + 13b²: Other remaining primes congruent to {1, 7, 9, 13, 15, 17, 19, 23, 25, 29, 33, 41, 49, 55, 59, 61, 63, 79, 81, 83, 91, 93, 97, 105, 107, 111, 117, 119, 121, 129, 131, 133, 135, 137, 139, 141, 149, 153, 155, 159, 161, 163, 167, 169, 171, 175, 179, 185, 195, 201, 203} mod 206, neither of the form a²+103b² nor 7a² + 6ab + 16b²

N can be written as a² + 103b² if and only if
- N is not congruent to 2 (mod 4).
- N has no prime factors congruent to {3, 5, 11, 21, 27, 31, 35, 37, 39, 43, 45, 47, 51, 53, 57, 65, 67, 69, 71, 73, 75, 77, 85, 87, 89, 95, 99, 101, 109, 113, 115, 123, 125, 127, 143, 145, 147, 151, 157, 165, 173, 177, 181, 183, 187, 189, 191, 193, 197, 199, 205} mod 206 to an odd power.
- If N is odd or N ≡ 4 (mod 8), then the (sum of exponents of prime factors of N of the form 7a² + 6ab + 16b², sum of exponents of prime factors of N of the form 8a² + 2ab + 13b²) is not (0,1) or (1,0) or (1,1) or (3,0) or (0,3).
- If N ≡ 8 (mod 16), then the (sum of exponents of prime factors of N of the form 7a² + 6ab + 16b², sum of exponents of prime factors of N of the form 8a² + 2ab + 13b²) is not (0,0) or (1,0) or (0,2).
- If N ≡ 16 (mod 32), then the (sum of exponents of prime factors of N of the form 7a² + 6ab + 16b², sum of exponents of prime factors of N of the form 8a² + 2ab + 13b²) is not (0,1).
- If N ≡ 32 (mod 64), then the (sum of exponents of prime factors of N of the form 7a² + 6ab + 16b², sum of exponents of prime factors of N of the form 8a² + 2ab + 13b²) is not (0,0).

k = 127

Primes p of the form a² + 127b²: Some primes congruent to {1, 9, 11, 13, 15, 17, 19, 21, 25, 31, 35, 37, 41, 47, 49, 61, 69, 71, 73, 79, 81, 87, 99, 103, 107, 113, 115, 117, 121, 127, 129, 131, 135, 143, 145, 149, 153, 157, 159, 161, 163, 165, 169, 171, 177, 179, 187, 189, 191, 195, 197, 199, 201, 203, 209, 211, 215, 221, 225, 227, 231, 247, 249, 251} mod 254, neither of the form 11a² + 8ab + 13b² nor 8a² + 6ab + 17b²
if N is a non-negative integer that can be written as a²+127b², then p × N can be written as a²+127b²
if N is a non-negative integer that cannot be written as a²+127b², then p × N cannot be written as a²+127b²
Primes of the form 11a² + 8ab + 13b²: Some more primes congruent to {1, 9, 11, 13, 15, 17, 19, 21, 25, 31, 35, 37, 41, 47, 49, 61, 69, 71, 73, 79, 81, 87, 99, 103, 107, 113, 115, 117, 121, 129, 131, 135, 143, 145, 149, 153, 157, 159, 161, 163, 165, 169, 171, 177, 179, 187, 189, 191, 195, 197, 199, 201, 203, 209, 211, 215, 221, 225, 227, 231, 247, 249, 251} mod 254, neither of the form a²+127b² nor 8a² + 6ab + 17b²
Primes of the form 8a² + 6ab + 17b²: Other remaining primes congruent to {1, 9, 11, 13, 15, 17, 19, 21, 25, 31, 35, 37, 41, 47, 49, 61, 69, 71, 73, 79, 81, 87, 99, 103, 107, 113, 115, 117, 121, 129, 131, 135, 143, 145, 149, 153, 157, 159, 161, 163, 165, 169, 171, 177, 179, 187, 189, 191, 195, 197, 199, 201, 203, 209, 211, 215, 221, 225, 227, 231, 247, 249, 251} mod 254, neither of the form a²+127b² nor 11a² + 8ab + 13b²

N can be written as a² + 127b² if and only if
- N is not congruent to 2 (mod 4).
- N has no prime factors congruent to {3, 5, 7, 23, 27, 29, 33, 39, 43, 45, 51, 53, 55, 57, 59, 63, 65, 67, 75, 77, 83, 85, 89, 91, 93, 95, 97, 101, 105, 109, 111, 119, 123, 125, 133, 137, 139, 141, 147, 151, 155, 167, 173, 175, 181, 183, 185, 193, 205, 207, 213, 217, 219, 223, 229, 233, 235, 237, 239, 241, 243, 245, 253} mod 254 to an odd power.
- If N is odd or N ≡ 4 (mod 8), then the (sum of exponents of prime factors of N of the form 11a² + 8ab + 13b², sum of exponents of prime factors of N of the form 8a² + 6ab + 17b²) is not (0,1) or (1,0) or (1,1) or (3,0) or (0,3).
- If N ≡ 8 (mod 16), then the (sum of exponents of prime factors of N of the form 11a² + 8ab + 13b², sum of exponents of prime factors of N of the form 8a² + 6ab + 17b²) is not (0,0) or (1,0) or (0,2).
- If N ≡ 16 (mod 32), then the (sum of exponents of prime factors of N of the form 11a² + 8ab + 13b², sum of exponents of prime factors of N of the form 8a² + 6ab + 17b²) is not (0,1).
- If N ≡ 32 (mod 64), then the (sum of exponents of prime factors of N of the form 11a² + 8ab + 13b², sum of exponents of prime factors of N of the form 8a² + 6ab + 17b²) is not (0,0).

Raman · 2013-03-21, 19:29

k = 71

Primes p of the form a² + 71b²: Some primes congruent to {1, 3, 5, 9, 15, 19, 25, 27, 29, 37, 43, 45, 49, 57, 71, 73, 75, 77, 79, 81, 83, 87, 89, 91, 95, 101, 103, 107, 109, 111, 119, 121, 125, 129, 131, 135} mod 142, neither of the form 3a² + 2ab + 24b² nor 5a² + 4ab + 15b² nor 8a² + 2ab + 9b²
if N is a non-negative integer that can be written as a²+71b², then p × N can be written as a²+71b²
if N is a non-negative integer that cannot be written as a²+71b², then p × N cannot be written as a²+71b²
Primes of the form 3a² + 2ab + 24b²: Some more primes congruent to {1, 3, 5, 9, 15, 19, 25, 27, 29, 37, 43, 45, 49, 57, 73, 75, 77, 79, 81, 83, 87, 89, 91, 95, 101, 103, 107, 109, 111, 119, 121, 125, 129, 131, 135} mod 142, neither of the form a²+71b² nor 5a² + 4ab + 15b² nor 8a² + 2ab + 9b²
Primes of the form 5a² + 4ab + 15b²: Some more primes congruent to {1, 3, 5, 9, 15, 19, 25, 27, 29, 37, 43, 45, 49, 57, 73, 75, 77, 79, 81, 83, 87, 89, 91, 95, 101, 103, 107, 109, 111, 119, 121, 125, 129, 131, 135} mod 142, neither of the form a²+71b² nor 3a² + 2ab + 24b² nor 8a² + 2ab + 9b²
Primes of the form 8a² + 2ab + 9b²: Other remaining primes congruent to {1, 3, 5, 9, 15, 19, 25, 27, 29, 37, 43, 45, 49, 57, 73, 75, 77, 79, 81, 83, 87, 89, 91, 95, 101, 103, 107, 109, 111, 119, 121, 125, 129, 131, 135} mod 142, neither of the form a²+71b² nor 3a² + 2ab + 24b² nor 5a² + 4ab + 15b²

N can be written as a² + 71b² if and only if
- N is not congruent to 2 (mod 4).
- N has no prime factors congruent to {7, 11, 13, 17, 21, 23, 31, 33, 35, 39, 41, 47, 51, 53, 55, 59, 61, 63, 65, 67, 69, 85, 93, 97, 99, 105, 113, 115, 117, 123, 127, 133, 137, 139, 141} mod 142 to an odd power.
- If N is odd or N ≡ 4 (mod 8), then the (sum of exponents of prime factors of N of the form 3a² + 2ab + 24b², sum of exponents of prime factors of N of the form 5a² + 4ab + 15b², sum of exponents of prime factors of N of the form 8a² + 2ab + 9b²) is not (1,0,0) or (0,1,0) or (0,0,1) or (1,1,0) or (1,0,1) or (0,1,1) or (3,0,0) or (0,3,0) or (0,0,3) or (2,1,0) or (0,2,1) or (1,0,2) or (1,3,0) or (0,1,3) or (3,0,1) or (5,0,0) or (0,5,0).
- If N ≡ 8 (mod 16), then the (sum of exponents of prime factors of N of the form 3a² + 2ab + 24b², sum of exponents of prime factors of N of the form 5a² + 4ab + 15b², sum of exponents of prime factors of N of the form 8a² + 2ab + 9b²) is not (0,0,0) or (1,0,0) or (0,1,0) or (1,0,1) or (0,2,0) or (0,0,2) or (0,1,2) or (3,0,0).
- If N ≡ 16 (mod 32), then the (sum of exponents of prime factors of N of the form 3a² + 2ab + 24b², sum of exponents of prime factors of N of the form 5a² + 4ab + 15b², sum of exponents of prime factors of N of the form 8a² + 2ab + 9b²) is not (1,0,0) or (0,0,1) or (0,1,1) or (0,0,3).
- If N ≡ 32 (mod 64), then the (sum of exponents of prime factors of N of the form 3a² + 2ab + 24b², sum of exponents of prime factors of N of the form 5a² + 4ab + 15b², sum of exponents of prime factors of N of the form 8a² + 2ab + 9b²) is not (0,0,0) or (0,1,0) or (0,0,2).
- If N ≡ 64 (mod 128), then the (sum of exponents of prime factors of N of the form 3a² + 2ab + 24b², sum of exponents of prime factors of N of the form 5a² + 4ab + 15b², sum of exponents of prime factors of N of the form 8a² + 2ab + 9b²) is not (0,0,1).
- If N ≡ 128 (mod 256), then the (sum of exponents of prime factors of N of the form 3a² + 2ab + 24b², sum of exponents of prime factors of N of the form 5a² + 4ab + 15b², sum of exponents of prime factors of N of the form 8a² + 2ab + 9b²) is not (0,0,0).

k = 115

Primes p of the form a² + 115b²: Some primes congruent to {1, 4, 6, 9, 16, 24, 26, 29, 31, 36, 39, 41, 49, 54, 59, 64, 71, 81, 94, 96, 101, 104} mod 115, not of the form 4a² + 2ab + 29b²
if N is a non-negative integer that can be written as a²+115b², then p × N can be written as a²+115b²
if N is a non-negative integer that cannot be written as a²+115b², then p × N cannot be written as a²+115b²
Primes of the form 4a² + 2ab + 29b²: Other remaining primes congruent to {1, 4, 6, 9, 16, 24, 26, 29, 31, 36, 39, 41, 49, 54, 59, 64, 71, 81, 94, 96, 101, 104} mod 115, not of the form a²+115b²
Primes of the form 7a² + 4ab + 17b²: Some primes congruent to {7, 17, 22, 28, 33, 37, 38, 42, 43, 53, 57, 63, 67, 68, 83, 88, 97, 102, 103, 107, 112, 113} mod 115, not of the form 5a² + 23b²
Primes of the form 5a² + 23b²: Other remaining primes congruent to {5, 7, 17, 22, 23, 28, 33, 37, 38, 42, 43, 53, 57, 63, 67, 68, 83, 88, 97, 102, 103, 107, 112, 113} mod 115, not of the form 7a² + 4ab + 17b²

N can be written as a² + 115b² if and only if
- N has no prime factors congruent to {2, 3, 8, 11, 12, 13, 14, 18, 19, 21, 27, 32, 34, 44, 47, 48, 51, 52, 56, 58, 61, 62, 66, 72, 73, 74, 76, 77, 78, 79, 82, 84, 86, 87, 89, 91, 93, 98, 99, 106, 108, 109, 111, 114} mod 115 to an odd power.
- Sum of exponents of prime factors of N of the form 7a² + 4ab + 17b² and the sum of exponents of prime factors of N of the form 5a² + 23b² are of the same parity.
- If N is odd, then the sum of exponents of prime factors of N of the form 7a² + 4ab + 17b² and the sum of exponents of prime factors of N of the form 4a² + 2ab + 29b² do not add up to one.

k = 123

Primes p of the form a² + 123b²: Some primes congruent to {1, 4, 10, 16, 25, 31, 37, 40, 43, 46, 49, 61, 64, 73, 91, 100, 103, 115, 118, 121} mod 123, not of the form 4a² + 2ab + 31b²
if N is a non-negative integer that can be written as a²+123b², then p × N can be written as a²+123b²
if N is a non-negative integer that cannot be written as a²+123b², then p × N cannot be written as a²+123b²
Primes of the form 4a² + 2ab + 31b²: Other remaining primes congruent to {1, 4, 10, 16, 25, 31, 37, 40, 43, 46, 49, 61, 64, 73, 91, 100, 103, 115, 118, 121} mod 123, not of the form a²+123b²
Primes of the form 11a² + 6ab + 12b²: Some primes congruent to {11, 14, 17, 26, 29, 35, 38, 44, 47, 53, 56, 65, 68, 71, 89, 95, 101, 104, 110, 116} mod 123, not of the form 3a² + 41b²
Primes of the form 3a² + 41b²: Other remaining primes congruent to {3, 11, 14, 17, 26, 29, 35, 38, 41, 44, 47, 53, 56, 65, 68, 71, 89, 95, 101, 104, 110, 116} mod 123, not of the form 11a² + 6ab + 12b²

N can be written as a² + 123b² if and only if
- N has no prime factors congruent to {2, 5, 7, 8, 13, 19, 20, 22, 23, 28, 32, 34, 50, 52, 55, 58, 59, 62, 67, 70, 74, 76, 77, 79, 80, 83, 85, 86, 88, 92, 94, 97, 98, 106, 107, 109, 112, 113, 119, 122} mod 123 to an odd power.
- Sum of exponents of prime factors of N of the form 11a² + 6ab + 12b² and the sum of exponents of prime factors of N of the form 3a² + 41b² are of the same parity.
- If N is odd, then the sum of exponents of prime factors of N of the form 11a² + 6ab + 12b² and the sum of exponents of prime factors of N of the form 4a² + 2ab + 31b² do not add up to one.

k = 124

Primes p of the form a² + 124b²: Some primes congruent to {1, 5, 9, 25, 33, 41, 45, 49, 69, 81, 97, 101, 109, 113, 121} mod 124, not of the form 5a² + 2ab + 25b²
if N is a non-negative integer that can be written as a²+124b², then p × N can be written as a²+124b²
if N is a non-negative integer that cannot be written as a²+124b², then p × N cannot be written as a²+124b²
Primes of the form 5a² + 2ab + 25b²: Other remaining primes congruent to {1, 5, 9, 25, 33, 41, 45, 49, 69, 81, 97, 101, 109, 113, 121} mod 124, not of the form a²+124b²
Primes of the form 7a² + 6ab + 19b²: Some primes congruent to {7, 19, 35, 39, 47, 51, 59, 63, 67, 71, 87, 95, 103, 107, 111} mod 124, not of the form 4a² + 31b²
Primes of the form 4a² + 31b²: Other remaining primes congruent to {7, 19, 31, 35, 39, 47, 51, 59, 63, 67, 71, 87, 95, 103, 107, 111} mod 124, not of the form 7a² + 6ab + 19b²

N can be written as a² + 124b² if and only if
- N is not congruent to 2 (mod 4) or 8 (mod 16).
- N has no prime factors congruent to {3, 11, 13, 15, 17, 21, 23, 27, 29, 37, 43, 53, 55, 57, 61, 65, 73, 75, 77, 79, 83, 85, 89, 91, 99, 105, 115, 117, 119, 123} mod 124 to an odd power.
- If N is odd, then the sum of exponents of prime factors of N of the form 7a² + 6ab + 19b² and the sum of exponents of prime factors of N of the form 4a² + 31b² are of the same parity.
- If N is odd or N ≡ 4 (mod 8) or N ≡ 16 (mod 32), then the sum of exponents of prime factors of N of the form 7a² + 6ab + 19b² and the sum of exponents of prime factors of N of the form 5a² + 2ab + 25b² do not add up to one.
- If N ≡ 32 (mod 64), then there is atleast one prime factor of N of the form 7a² + 6ab + 19b² or 5a² + 2ab + 25b².

Raman · 2013-03-22, 19:19

k = 106

Primes p of the form a² + 106b²: Some primes congruent to {1, 9, 11, 17, 25, 43, 49, 57, 59, 81, 89, 91, 97, 99, 105, 107, 113, 115, 121, 123, 131, 153, 155, 163, 169, 187, 195, 201, 203, 211, 219, 225, 227, 241, 249, 259, 275, 281, 289, 305, 307, 329, 331, 347, 355, 361, 377, 387, 395, 409, 411, 417} mod 424, not of the form 10a² + 4ab + 11b²
if N is a non-negative integer that can be written as a²+106b², then p × N can be written as a²+106b²
if N is a non-negative integer that cannot be written as a²+106b², then p × N cannot be written as a²+106b²
Primes of the form 10a² + 4ab + 11b²: Other remaining primes congruent to {1, 9, 11, 17, 25, 43, 49, 57, 59, 81, 89, 91, 97, 99, 105, 107, 113, 115, 121, 123, 131, 153, 155, 163, 169, 187, 195, 201, 203, 211, 219, 225, 227, 241, 249, 259, 275, 281, 289, 305, 307, 329, 331, 347, 355, 361, 377, 387, 395, 409, 411, 417} mod 424, not of the form a²+106b²
Primes of the form 5a² + 4ab + 22b²: Some primes congruent to {5, 21, 23, 31, 39, 45, 55, 61, 71, 79, 85, 87, 101, 103, 109, 111, 125, 127, 133, 141, 151, 157, 167, 173, 181, 189, 191, 207, 215, 231, 239, 245, 247, 253, 263, 277, 279, 285, 287, 295, 341, 349, 351, 357, 359, 373, 383, 389, 391, 397, 405, 421} mod 424, not of the form 2a² + 53b²
Primes of the form 2a² + 53b²: Other remaining primes congruent to {2, 5, 21, 23, 31, 39, 45, 53, 55, 61, 71, 79, 85, 87, 101, 103, 109, 111, 125, 127, 133, 141, 151, 157, 167, 173, 181, 189, 191, 207, 215, 231, 239, 245, 247, 253, 263, 277, 279, 285, 287, 295, 341, 349, 351, 357, 359, 373, 383, 389, 391, 397, 405, 421} mod 424, not of the form 5a² + 4ab + 22b²

N can be written as a² + 106b² if and only if
- N has no prime factors congruent to {3, 7, 13, 15, 19, 27, 29, 33, 35, 37, 41, 47, 51, 63, 65, 67, 69, 73, 75, 77, 83, 93, 95, 117, 119, 129, 135, 137, 139, 143, 145, 147, 149, 161, 165, 171, 175, 177, 179, 183, 185, 193, 197, 199, 205, 209, 213, 217, 221, 223, 229, 233, 235, 237, 243, 251, 255, 257, 261, 267, 269, 271, 273, 283, 291, 293, 297, 299, 301, 303, 309, 311, 313, 315, 317, 319, 321, 323, 325, 327, 333, 335, 337, 339, 343, 345, 353, 363, 365, 367, 369, 375, 379, 381, 385, 393, 399, 401, 403, 407, 413, 415, 419, 423} mod 424 to an odd power.
- Sum of exponents of prime factors of N of the form 5a² + 4ab + 22b² and the sum of exponents of prime factors of N of the form 2a² + 53b² are of the same parity.
- Sum of exponents of prime factors of N of the form 5a² + 4ab + 22b² and the sum of exponents of prime factors of N of the form 10a² + 4ab + 11b² do not add up to one.

k = 108

Primes p of the form a² + 108b²: Some primes congruent to {1} mod 12, not of the form 9a² + 6ab + 13b²
if N is a non-negative integer that can be written as a²+108b², then p × N can be written as a²+108b²
if N is a non-negative integer that cannot be written as a²+108b², then p × N cannot be written as a²+108b²
Primes of the form 9a² + 6ab + 13b²: Other remaining primes congruent to {1} mod 12, not of the form a²+108b²
Primes of the form 7a² + 4ab + 16b²: Some primes congruent to {7} mod 12, not of the form 4a² + 27b²
Primes of the form 4a² + 27b²: Other remaining primes congruent to {7} mod 12, not of the form 7a² + 4ab + 16b²

N can be written as a² + 108b² if and only if
- N is not congruent to {3, 6} mod 9.
- N has no prime factors congruent to {2, 5, 11} mod 12 to an odd power.
- If N is odd, then the sum of exponents of {3, 7} mod 12 prime factors of N is even.
- If N ≡ {1, 5} mod 6, or if N ≡ {4, 20} mod 24, then the sum of exponents of prime factors of N of the form 7a² + 4ab + 16b² and the sum of exponents of prime factors of N of the form 9a² + 6ab + 13b² do not add up to one.

k = 109

Primes p of the form a² + 109b²: Some primes congruent to {1, 5, 9, 21, 25, 29, 45, 49, 61, 73, 81, 89, 93, 97, 105, 109, 113, 121, 125, 129, 137, 145, 157, 169, 173, 189, 193, 197, 209, 213, 217, 221, 225, 233, 245, 249, 253, 261, 281, 289, 293, 301, 305, 349, 353, 361, 365, 373, 393, 401, 405, 409, 421, 429, 433} mod 436, not of the form 5a² + 2ab + 22b²
if N is a non-negative integer that can be written as a²+109b², then p × N can be written as a²+109b²
if N is a non-negative integer that cannot be written as a²+109b², then p × N cannot be written as a²+109b²
Primes of the form 5a² + 2ab + 22b²: Other remaining primes congruent to {1, 5, 9, 21, 25, 29, 45, 49, 61, 73, 81, 89, 93, 97, 105, 113, 121, 125, 129, 137, 145, 157, 169, 173, 189, 193, 197, 209, 213, 217, 221, 225, 233, 245, 249, 253, 261, 281, 289, 293, 301, 305, 349, 353, 361, 365, 373, 393, 401, 405, 409, 421, 429, 433} mod 436, not of the form a²+109b²
Primes of the form 10a² + 2ab + 11b²: Some primes congruent to {11, 19, 23, 39, 47, 51, 55, 59, 67, 79, 91, 95, 99, 103, 107, 111, 115, 119, 123, 127, 139, 151, 159, 163, 167, 171, 179, 195, 199, 207, 231, 235, 251, 255, 259, 271, 275, 283, 287, 295, 303, 319, 335, 351, 359, 367, 371, 379, 383, 395, 399, 403, 419, 423} mod 436, not of the form 2a² + 2ab + 55b²
Primes of the form 2a² + 2ab + 55b²: Other remaining primes congruent to {2, 11, 19, 23, 39, 47, 51, 55, 59, 67, 79, 91, 95, 99, 103, 107, 111, 115, 119, 123, 127, 139, 151, 159, 163, 167, 171, 179, 195, 199, 207, 231, 235, 251, 255, 259, 271, 275, 283, 287, 295, 303, 319, 335, 351, 359, 367, 371, 379, 383, 395, 399, 403, 419, 423} mod 436, not of the form 10a² + 2ab + 11b²

N can be written as a² + 109b² if and only if
- N has no prime factors congruent to {3, 7, 13, 15, 17, 27, 31, 33, 35, 37, 41, 43, 53, 57, 63, 65, 69, 71, 75, 77, 83, 85, 87, 101, 117, 131, 133, 135, 141, 143, 147, 149, 153, 155, 161, 165, 175, 177, 181, 183, 185, 187, 191, 201, 203, 205, 211, 215, 219, 223, 227, 229, 237, 239, 241, 243, 247, 257, 263, 265, 267, 269, 273, 277, 279, 285, 291, 297, 299, 307, 309, 311, 313, 315, 317, 321, 323, 325, 329, 331, 333, 337, 339, 341, 343, 345, 347, 355, 357, 363, 369, 375, 377, 381, 385, 387, 389, 391, 397, 407, 411, 413, 415, 417, 425, 427, 431, 435} mod 436 to an odd power.
- Sum of exponents of prime factors of N of the form 10a² + 2ab + 11b² and the sum of exponents of prime factors of N of the form 2a² + 2ab + 55b² are of the same parity.
- Sum of exponents of prime factors of N of the form 10a² + 2ab + 11b² and the sum of exponents of prime factors of N of the form 5a² + 2ab + 22b² do not add up to one.

k = 118

Primes p of the form a² + 118b²: Some primes congruent to {1, 7, 9, 15, 17, 25, 41, 49, 57, 63, 71, 79, 81, 87, 95, 105, 119, 121, 127, 135, 137, 143, 145, 153, 159, 167, 169, 175, 193, 199, 223, 225, 239, 241, 255, 257, 263, 265, 271, 281, 287, 289, 311, 321, 343, 359, 361, 369, 375, 383, 399, 407, 417, 425, 433, 439, 441, 449} mod 472, not of the form 7a² + 2ab + 17b²
if N is a non-negative integer that can be written as a²+118b², then p × N can be written as a²+118b²
if N is a non-negative integer that cannot be written as a²+118b², then p × N cannot be written as a²+118b²
Primes of the form 7a² + 2ab + 17b²: Other remaining primes congruent to {1, 7, 9, 15, 17, 25, 41, 49, 57, 63, 71, 79, 81, 87, 95, 105, 119, 121, 127, 135, 137, 143, 145, 153, 159, 167, 169, 175, 193, 199, 223, 225, 239, 241, 255, 257, 263, 265, 271, 281, 287, 289, 311, 321, 343, 359, 361, 369, 375, 383, 399, 407, 417, 425, 433, 439, 441, 449} mod 472, not of the form a²+118b²
Primes of the form 11a² + 10ab + 13b²: Some primes congruent to {11, 13, 37, 43, 61, 67, 69, 77, 83, 91, 93, 99, 101, 109, 115, 117, 131, 141, 149, 155, 157, 165, 173, 179, 187, 195, 211, 219, 221, 227, 229, 235, 259, 267, 269, 275, 283, 291, 301, 309, 325, 333, 339, 347, 349, 365, 387, 397, 419, 421, 427, 437, 443, 445, 451, 453, 467, 469} mod 472, not of the form 2a² + 59b²
Primes of the form 2a² + 59b²: Other remaining primes congruent to {2, 11, 13, 37, 43, 59, 61, 67, 69, 77, 83, 91, 93, 99, 101, 109, 115, 117, 131, 141, 149, 155, 157, 165, 173, 179, 187, 195, 211, 219, 221, 227, 229, 235, 259, 267, 269, 275, 283, 291, 301, 309, 325, 333, 339, 347, 349, 365, 387, 397, 419, 421, 427, 437, 443, 445, 451, 453, 467, 469} mod 472, not of the form 11a² + 10ab + 13b²

N can be written as a² + 118b² if and only if
- N has no prime factors congruent to {3, 5, 19, 21, 23, 27, 29, 31, 33, 35, 39, 45, 47, 51, 53, 55, 65, 73, 75, 85, 89, 97, 103, 107, 111, 113, 123, 125, 129, 133, 139, 147, 151, 161, 163, 171, 181, 183, 185, 189, 191, 197, 201, 203, 205, 207, 209, 213, 215, 217, 231, 233, 237, 243, 245, 247, 249, 251, 253, 261, 273, 277, 279, 285, 293, 297, 299, 303, 305, 307, 313, 315, 317, 319, 323, 327, 329, 331, 335, 337, 341, 345, 351, 353, 355, 357, 363, 367, 371, 373, 377, 379, 381, 385, 389, 391, 393, 395, 401, 403, 405, 409, 411, 415, 423, 429, 431, 435, 447, 455, 457, 459, 461, 463, 465, 471} mod 472 to an odd power.
- Sum of exponents of prime factors of N of the form 11a² + 10ab + 13b² and the sum of exponents of prime factors of N of the form 2a² + 59b² are of the same parity.
- If N is odd, then the sum of exponents of prime factors of N of the form 11a² + 10ab + 13b² and the sum of exponents of prime factors of N of the form 7a² + 2ab + 17b² do not add up to one.

Raman · 2013-03-22, 21:43

k = 121

Primes p of the form a² + 121b²: Some primes congruent to {1, 5, 9, 25, 37} mod 44, not of the form 5a² + 4ab + 25b²
if N is a non-negative integer that can be written as a²+121b², then p × N can be written as a²+121b²
if N is a non-negative integer that cannot be written as a²+121b², then p × N cannot be written as a²+121b²
Primes of the form 5a² + 4ab + 25b²: Other remaining primes congruent to {1, 5, 9, 25, 37} mod 44, not of the form a²+121b²
Primes of the form 10a² + 6ab + 13b²: Some primes congruent to {13, 17, 21, 29, 41} mod 44, not of the form 2a² + 2ab + 61b²
Primes of the form 2a² + 2ab + 61b²: Other remaining primes congruent to {2, 13, 17, 21, 29, 41} mod 44, not of the form 10a² + 6ab + 13b²

N can be written as a² + 121b² if and only if
- N has no prime factors congruent to {3, 7, 11, 15, 19, 23, 27, 31, 35, 39, 43} mod 44 to an odd power.
- If N is not a multiple of 11, then the sum of exponents of prime factors of N of the form 10a² + 6ab + 13b² and the sum of exponents of prime factors of N of the form 2a² + 2ab + 61b² are of the same parity.
- If N is not a multiple of 11, then the sum of exponents of prime factors of N of the form 10a² + 6ab + 13b² and the sum of exponents of prime factors of N of the form 5a² + 4ab + 25b² do not add up to one.

k = 135

Primes p of the form a² + 135b²: Some primes congruent to {1, 19} mod 30, not of the form 9a² + 12ab + 19b²
if N is a non-negative integer that can be written as a²+135b², then p × N can be written as a²+135b²
if N is a non-negative integer that cannot be written as a²+135b², then p × N cannot be written as a²+135b²
Primes of the form 9a² + 12ab + 19b²: Other remaining primes congruent to {1, 19} mod 30, not of the form a²+135b²
Primes of the form 8a² + 2ab + 17b²: Some primes congruent to {17, 23} mod 30, not of the form 5a² + 27b²
Primes of the form 5a² + 27b²: Other remaining primes congruent to {5, 17, 23} mod 30, not of the form 8a² + 2ab + 17b²

N can be written as a² + 135b² if and only if
- N is not congruent to 2 (mod 4).
- N is not congruent to {3, 6} (mod 9).
- N has no prime factors congruent to {7, 11, 13, 29} mod 30 to an odd power.
- Sum of exponents of {2, 3, 5, 17, 23} mod 30 prime factors of N is even.
- If N ≡ {1, 5} mod 6, or if N ≡ {4, 20} mod 24, then the sum of exponents of prime factors of N of the form 8a² + 2ab + 17b² and the sum of exponents of prime factors of N of the form 9a² + 12ab + 19b² do not add up to one.
- If N ≡ {8, 40} mod 48, then there is atleast one prime factor of N of the form 8a² + 2ab + 17b² or 9a² + 12ab + 19b².

k = 142

Primes p of the form a² + 142b²: Some primes congruent to {1, 9, 15, 25, 49, 57, 73, 79, 81, 87, 89, 95, 103, 111, 119, 121, 129, 135, 143, 145, 151, 161, 167, 169, 185, 191, 199, 215, 217, 223, 225, 231, 233, 249, 263, 271, 273, 287, 289, 303, 311, 313, 321, 327, 329, 359, 361, 367, 375, 385, 391, 393, 409, 415, 431, 441, 455, 463, 471, 503, 505, 513, 521, 527, 529, 535, 537, 545, 551, 561} mod 568, not of the form 2a² + 71b²
if N is a non-negative integer that can be written as a²+142b², then p × N can be written as a²+142b²
if N is a non-negative integer that cannot be written as a²+142b², then p × N cannot be written as a²+142b²
Primes of the form 2a² + 71b²: Other remaining primes congruent to {1, 2, 9, 15, 25, 49, 57, 71, 73, 79, 81, 87, 89, 95, 103, 111, 119, 121, 129, 135, 143, 145, 151, 161, 167, 169, 185, 191, 199, 215, 217, 223, 225, 231, 233, 249, 263, 271, 273, 287, 289, 303, 311, 313, 321, 327, 329, 359, 361, 367, 375, 385, 391, 393, 409, 415, 431, 441, 455, 463, 471, 503, 505, 513, 521, 527, 529, 535, 537, 545, 551, 561} mod 568, not of the form a²+142b²
Primes of the form 11a² + 2ab + 13b²: All primes congruent to {11, 13, 21, 35, 51, 53, 59, 61, 67, 69, 85, 93, 99, 115, 117, 123, 133, 139, 141, 149, 155, 163, 165, 173, 181, 189, 195, 197, 203, 205, 211, 227, 235, 259, 269, 275, 283, 291, 301, 307, 315, 317, 323, 325, 331, 339, 347, 349, 381, 389, 397, 411, 421, 437, 443, 459, 461, 467, 477, 485, 491, 493, 523, 525, 531, 539, 541, 549, 563, 565} mod 568

N can be written as a² + 142b² if and only if
- N has no prime factors congruent to {3, 5, 7, 17, 19, 23, 27, 29, 31, 33, 37, 39, 41, 43, 45, 47, 55, 63, 65, 75, 77, 83, 91, 97, 101, 105, 107, 109, 113, 125, 127, 131, 137, 147, 153, 157, 159, 171, 175, 177, 179, 183, 187, 193, 201, 207, 209, 219, 221, 229, 237, 239, 241, 243, 245, 247, 251, 253, 255, 257, 261, 265, 267, 277, 279, 281, 285, 293, 295, 297, 299, 305, 309, 319, 333, 335, 337, 341, 343, 345, 351, 353, 357, 363, 365, 369, 371, 373, 377, 379, 383, 387, 395, 399, 401, 403, 405, 407, 413, 417, 419, 423, 425, 427, 429, 433, 435, 439, 445, 447, 449, 451, 453, 457, 465, 469, 473, 475, 479, 481, 483, 487, 489, 495, 499, 501, 507, 509, 511, 515, 517, 519, 533, 543, 547, 553, 555, 557, 559, 567} mod 568 to an odd power.
- Sum of exponents of {11, 13, 21, 35, 51, 53, 59, 61, 67, 69, 85, 93, 99, 115, 117, 123, 133, 139, 141, 149, 155, 163, 165, 173, 181, 189, 195, 197, 203, 205, 211, 227, 235, 259, 269, 275, 283, 291, 301, 307, 315, 317, 323, 325, 331, 339, 347, 349, 381, 389, 397, 411, 421, 437, 443, 459, 461, 467, 477, 485, 491, 493, 523, 525, 531, 539, 541, 549, 563, 565} mod 568 prime factors of N is even.
- If N has no prime factors congruent to {11, 13, 21, 35, 51, 53, 59, 61, 67, 69, 85, 93, 99, 115, 117, 123, 133, 139, 141, 149, 155, 163, 165, 173, 181, 189, 195, 197, 203, 205, 211, 227, 235, 259, 269, 275, 283, 291, 301, 307, 315, 317, 323, 325, 331, 339, 347, 349, 381, 389, 397, 411, 421, 437, 443, 459, 461, 467, 477, 485, 491, 493, 523, 525, 531, 539, 541, 549, 563, 565} mod 568, then sum of exponents of prime factors of N of the form 2a² + 71b² is even.

k = 147

Primes p of the form a² + 147b²: Some primes congruent to {1, 16, 25} mod 21, not of the form 4a² + 2ab + 37b²
if N is a non-negative integer that can be written as a²+147b², then p × N can be written as a²+147b²
if N is a non-negative integer that cannot be written as a²+147b², then p × N cannot be written as a²+147b²
Primes of the form 4a² + 2ab + 37b²: Other remaining primes congruent to {1, 16, 25} mod 21, not of the form a²+147b²
Primes of the form 12a² + 6ab + 13b²: Some primes congruent to {10, 13, 19} mod 21, not of the form 3a² + 49b²
Primes of the form 3a² + 49b²: Other remaining primes congruent to {3, 10, 13, 19} mod 21, not of the form 12a² + 6ab + 13b²

N can be written as a² + 147b² if and only if
- N is not congruent to {7, 14, 21, 28, 35, 42} (mod 49).
- N has no prime factors congruent to {2, 5, 8, 11, 17, 20} mod 21 to an odd power.
- If N is not a multiple of 7, then the sum of exponents of prime factors of N of the form 12a² + 6ab + 13b² and the sum of exponents of prime factors of N of the form 3a² + 49b² are of the same parity.
- If N is co-prime to 14, then the sum of exponents of prime factors of N of the form 12a² + 6ab + 13b² and the sum of exponents of prime factors of N of the form 4a² + 2ab + 37b² do not add up to one.

k = 148

Primes p of the form a² + 148b²: Some primes congruent to {1, 9, 21, 25, 33, 41, 49, 53, 65, 73, 77, 81, 85, 101, 121, 137, 141, 145} mod 148, not of the form 4a² + 37b²
if N is a non-negative integer that can be written as a²+148b², then p × N can be written as a²+148b²
if N is a non-negative integer that cannot be written as a²+148b², then p × N cannot be written as a²+148b²
Primes of the form 4a² + 37b²: Other remaining primes congruent to {1, 9, 21, 25, 33, 37, 41, 49, 53, 65, 73, 77, 81, 85, 101, 121, 137, 141, 145} mod 148, not of the form a²+148b²
Primes of the form 8a² + 4ab + 19b²: All primes congruent to {15, 19, 23, 31, 35, 39, 43, 51, 55, 59, 79, 87, 91, 103, 119, 131, 135, 143} mod 148

N can be written as a² + 148b² if and only if
- N is not congruent to 2 (mod 4).
- N has no prime factors congruent to {3, 5, 7, 11, 13, 17, 27, 29, 45, 47, 57, 61, 63, 67, 69, 71, 75, 83, 89, 93, 95, 97, 99, 105, 107, 109, 113, 115, 117, 123, 125, 127, 129, 133, 139, 147} mod 148 to an odd power.
- Sum of exponents of {2, 15, 19, 23, 31, 35, 39, 43, 51, 55, 59, 79, 87, 91, 103, 119, 131, 135, 143} mod 148 prime factors of N is even.
- If N has no prime factors congruent to {2, 15, 19, 23, 31, 35, 39, 43, 51, 55, 59, 79, 87, 91, 103, 119, 131, 135, 143} mod 148, then sum of exponents of prime factors of N of the form 4a² + 37b² is even.

Raman · 2013-03-22, 22:24

Quote:

Originally Posted by Raman

$\underline{Claim\ VI} \\ No\ prime\ number\ p\ can\ be\ generated\ by\ using\ two\ different\ quadratic\ form\ polynomials\ of\ same\ discriminant\ =\ -4k$

$\underline{Claim\ IX} \\ Any\ prime\ number\ p\ for\ which\ -k\ is\ being\ a\ quadratic\ residue\ (mod\ p)\ is\ generated\ by\ using\ exactly\ one\ inequivalent\ quadratic\ form\ polynomial\ of\ discriminant\ =\ -4k\ \\ and\ then\ for\ all\ these\ prime\ numbers\ p\ there\ is\ being\ \strike{the}\ some\ multiple\ of\ p,\ to\ which\ p\ is\ being\ raised\ to\ an\ odd\ power,\ that\ can\ be\ written\ as\ a^2+kb^2$

$\underline{Claim\ XII} \\ No\ quadratic\ form\ polynomial\ of\ discriminant\ =\ -4k\ can\ be\ able\ to\ generate\ a\ prime\ number\ p,\ for\ which\ -k\ is\ being\ a\ quadratic\ non-residue\ (mod\ p) \\ and\ then\ for\ all\ these\ prime\ numbers\ p\ there\ is\ being\ \strike{the}\ no\ multiple\ of\ p,\ to\ which\ p\ is\ being\ raised\ to\ an\ odd\ power,\ that\ can\ be\ written\ as\ a^2+kb^2$

$Certain\ set\ of\ residue\ classes\ (mod\ 4k)\ is\ being\ the\ modulo\ usually\ being\ used\ in\ order\ to\ classify\ /\ categorize\ the\ any\ prime\ numbers\ p,\ \\ but\ that\ which\ \strike{some\ same\ for\ the\ any\ way}\ are\ being\ generated\ by\ using\ some\ quadratic\ form\ polynomials\ of\ same\ discriminant\ =\ -4k$
$\white{\strike{r\ r\ of\ r\ the\ r\ r}}\$
[COLOR=White]and then certainly that always that
but that which[/COLOR]

$\underline{Claim\ XIII} \\ All\ prime\ numbers\ p\ for\ which\ -k\ is\ being\ a\ quadratic\ residue\ (mod\ p)\ can\ be\ generated\ by\ using\ a\ finite\ number\ of\ inequivalent\ quadratic\ form\ polynomials\ of\ discriminant\ =\ -4k\ \\ and\ then\ so\ lesser\ enough\ inequivalent\ quadratic\ form\ polynomials\ for\ those\ prime\ numbers\ p\ such\ that\ p\ =\ k\ or\ p \equiv x^2\ (mod\ 4k)\ or\ p \equiv x^2\ +\ k\ (mod\ 4k) \\ and\ then\ so\ lesser\ enough\ inequivalent\ quadratic\ form\ polynomials\ for\ those\ prime\ numbers\ p\ such\ that\ p\ \ne\ k\ and\ p \not\equiv x^2\ (mod\ 4k)\ and\ p \not\equiv x^2\ +\ k\ (mod\ 4k)\ \strike{such\ that}\ and\ then\ so\ \strike{for\ which}\ -k\ is\ being\ a\ quadratic\ residue\ (mod\ p) \\ and\ then\ for\ all\ these\ prime\ numbers\ p\ there\ is\ being\ \strike{the}\ some\ multiple\ of\ p,\ to\ which\ p\ is\ being\ raised\ to\ an\ odd\ power,\ that\ can\ be\ written\ as\ a^2+kb^2$ $\white{\strike{but\ that\ which\ some\ same\ for\ the\ any\ way}}\$ $\white{\strike{r\ r\ of\ r\ the\ r\ r}}\$

Raman · 2013-03-23, 09:44

Quote:

Originally Posted by Raman

$24 = 4! = \frac{385-1^2}{4^2} = \frac{385-13^2}{3^2} = \frac{385-17^2}{2^2} = \frac{385-19^2}{1^2} = 4! = 24$

385 - 24 × 1² = 19²
385 - 24 × 2² = 17²
385 - 24 × 3² = 13²
385 - 24 × 4² = 1²

385 = 5 × 7 × 11
as follows as
5 is being generated by using 5a²+2ab+5b² as form as following as a = 0, b = 1
7 is being generated by using 4a²+4ab+7b² as form as following as a = 0, b = 1
11 is being generated by using 3a²+6ab+11b² as form as following as a = 0, b = 1

all of them having got discriminant = -4k = -96 : as since as k = 24 as follows as

Raman · 2013-03-23, 13:48

Quote:

Originally Posted by Raman

$\underline{Claim\ VI} \\ No\ prime\ number\ p\ can\ be\ generated\ by\ using\ two\ different\ quadratic\ form\ polynomials\ of\ same\ discriminant\ =\ -4k$

$\underline{Claim\ IX} \\ Any\ prime\ number\ p\ for\ which\ -k\ is\ being\ a\ quadratic\ residue\ (mod\ p)\ is\ generated\ by\ using\ exactly\ one\ inequivalent\ quadratic\ form\ polynomial\ of\ discriminant\ =\ -4k\ \\ and\ then\ for\ all\ these\ prime\ numbers\ p\ there\ is\ being\ \strike{the}\ some\ multiple\ of\ p,\ to\ which\ p\ is\ being\ raised\ to\ an\ odd\ power,\ that\ can\ be\ written\ as\ a^2+kb^2$

$\underline{Claim\ XII} \\ No\ quadratic\ form\ polynomial\ of\ discriminant\ =\ -4k\ can\ be\ able\ to\ generate\ a\ prime\ number\ p,\ for\ which\ -k\ is\ being\ a\ quadratic\ non-residue\ (mod\ p) \\ and\ then\ for\ all\ these\ prime\ numbers\ p\ there\ is\ being\ \strike{the}\ no\ multiple\ of\ p,\ to\ which\ p\ is\ being\ raised\ to\ an\ odd\ power,\ that\ can\ be\ written\ as\ a^2+kb^2$

$Certain\ set\ of\ residue\ classes\ (mod\ 4k)\ is\ being\ the\ modulo\ usually\ being\ used\ in\ order\ to\ classify\ /\ categorize\ the\ any\ prime\ numbers\ p,\ \\ but\ that\ which\ \strike{some\ same\ for\ the\ any\ way}\ are\ being\ generated\ by\ using\ some\ quadratic\ form\ polynomials\ of\ same\ discriminant\ =\ -4k$

$\white{\strike{r\ r\ of\ r\ the\ r\ r}}\$

[COLOR=White]and then certainly that always that
but that which[/COLOR]

$\underline{Claim\ XIII} \\ All\ prime\ numbers\ p\ for\ which\ -k\ is\ being\ a\ quadratic\ residue\ (mod\ p)\ can\ be\ generated\ by\ using\ a\ finite\ number\ of\ inequivalent\ quadratic\ form\ polynomials\ of\ discriminant\ =\ -4k\ \\ and\ then\ so\ lesser\ enough\ inequivalent\ quadratic\ form\ polynomials\ for\ those\ prime\ numbers\ p\ such\ that\ p\ =\ k\ or\ p \equiv x^2\ (mod\ 4k)\ or\ p \equiv x^2\ +\ k\ (mod\ 4k) \\ and\ then\ so\ lesser\ enough\ inequivalent\ quadratic\ form\ polynomials\ for\ those\ prime\ numbers\ p\ such\ that\ p\ \ne\ k\ and\ p \not\equiv x^2\ (mod\ 4k)\ and\ p \not\equiv x^2\ +\ k\ (mod\ 4k)\ \strike{such\ that}\ and\ then\ so\ \strike{for\ which}\ -k\ is\ being\ a\ quadratic\ residue\ (mod\ p) \\ and\ then\ for\ all\ these\ prime\ numbers\ p\ there\ is\ being\ \strike{the}\ some\ multiple\ of\ p,\ to\ which\ p\ is\ being\ raised\ to\ an\ odd\ power,\ that\ can\ be\ written\ as\ a^2+kb^2$ $\white{\strike{but\ that\ which\ some\ same\ for\ the\ any\ way}}\$ $\white{\strike{r\ r\ of\ r\ the\ r\ r}}\$

Raman · 2013-03-23, 23:34

k = 151

Primes p of the form a² + 151b²: Some primes congruent to {1, 5, 9, 11, 17, 19, 21, 25, 29, 31, 37, 39, 43, 45, 47, 49, 55, 59, 69, 81, 85, 91, 95, 97, 99, 103, 105, 121, 123, 125, 127, 137, 139, 145, 151, 153, 155, 159, 161, 167, 169, 171, 173, 183, 185, 187, 189, 191, 193, 195, 201, 209, 213, 215, 219, 223, 225, 227, 229, 231, 235, 237, 239, 241, 245, 249, 251, 261, 267, 269, 275, 279, 287, 289, 295, 299} mod 302, neither of the form 5a² + 4ab + 31b² nor 11a² + 10ab + 16b² nor 8a² + 2ab + 19b²
if N is a non-negative integer that can be written as a²+151b², then p × N can be written as a²+151b²
if N is a non-negative integer that cannot be written as a²+151b², then p × N cannot be written as a²+151b²
Primes of the form 5a² + 4ab + 31b²: Some more primes congruent to {1, 5, 9, 11, 17, 19, 21, 25, 29, 31, 37, 39, 43, 45, 47, 49, 55, 59, 69, 81, 85, 91, 95, 97, 99, 103, 105, 121, 123, 125, 127, 137, 139, 145, 153, 155, 159, 161, 167, 169, 171, 173, 183, 185, 187, 189, 191, 193, 195, 201, 209, 213, 215, 219, 223, 225, 227, 229, 231, 235, 237, 239, 241, 245, 249, 251, 261, 267, 269, 275, 279, 287, 289, 295, 299} mod 302, neither of the form a²+151b² nor 11a² + 10ab + 16b² nor 8a² + 2ab + 19b²
Primes of the form 11a² + 10ab + 16b²: Some more primes congruent to {1, 5, 9, 11, 17, 19, 21, 25, 29, 31, 37, 39, 43, 45, 47, 49, 55, 59, 69, 81, 85, 91, 95, 97, 99, 103, 105, 121, 123, 125, 127, 137, 139, 145, 153, 155, 159, 161, 167, 169, 171, 173, 183, 185, 187, 189, 191, 193, 195, 201, 209, 213, 215, 219, 223, 225, 227, 229, 231, 235, 237, 239, 241, 245, 249, 251, 261, 267, 269, 275, 279, 287, 289, 295, 299} mod 302, neither of the form a²+151b² nor 5a² + 4ab + 31b² nor 8a² + 2ab + 19b²
Primes of the form 8a² + 2ab + 19b²: Other remaining primes congruent to {1, 5, 9, 11, 17, 19, 21, 25, 29, 31, 37, 39, 43, 45, 47, 49, 55, 59, 69, 81, 85, 91, 95, 97, 99, 103, 105, 121, 123, 125, 127, 137, 139, 145, 153, 155, 159, 161, 167, 169, 171, 173, 183, 185, 187, 189, 191, 193, 195, 201, 209, 213, 215, 219, 223, 225, 227, 229, 231, 235, 237, 239, 241, 245, 249, 251, 261, 267, 269, 275, 279, 287, 289, 295, 299} mod 302, neither of the form a²+151b² nor 5a² + 4ab + 31b² nor 11a² + 10ab + 16b²

N can be written as a² + 151b² if and only if
- N is not congruent to 2 (mod 4).
- N has no prime factors congruent to {3, 7, 13, 15, 23, 27, 33, 35, 41, 51, 53, 57, 61, 63, 65, 67, 71, 73, 75, 77, 79, 83, 87, 89, 93, 101, 107, 109, 111, 113, 115, 117, 119, 129, 131, 133, 135, 141, 143, 147, 149, 157, 163, 165, 175, 177, 179, 181, 197, 199, 203, 205, 207, 211, 217, 221, 233, 243, 247, 253, 255, 257, 259, 263, 265, 271, 273, 277, 281, 283, 285, 291, 293, 297, 301} mod 302 to an odd power.
- If N is odd or N ≡ 4 (mod 8), then the (sum of exponents of prime factors of N of the form 5a² + 4ab + 31b², sum of exponents of prime factors of N of the form 11a² + 10ab + 16b², sum of exponents of prime factors of N of the form 8a² + 2ab + 19b²) is not (1,0,0) or (0,1,0) or (0,0,1) or (1,1,0) or (1,0,1) or (0,1,1) or (3,0,0) or (0,3,0) or (0,0,3) or (2,1,0) or (0,2,1) or (1,0,2) or (1,3,0) or (0,1,3) or (3,0,1) or (5,0,0) or (0,5,0).
- If N ≡ 8 (mod 16), then the (sum of exponents of prime factors of N of the form 5a² + 4ab + 31b², sum of exponents of prime factors of N of the form 11a² + 10ab + 16b², sum of exponents of prime factors of N of the form 8a² + 2ab + 19b²) is not (0,0,0) or (1,0,0) or (0,1,0) or (1,0,1) or (0,2,0) or (0,0,2) or (0,1,2) or (3,0,0).
- If N ≡ 16 (mod 32), then the (sum of exponents of prime factors of N of the form 5a² + 4ab + 31b², sum of exponents of prime factors of N of the form 11a² + 10ab + 16b², sum of exponents of prime factors of N of the form 8a² + 2ab + 19b²) is not (1,0,0) or (0,0,1) or (0,1,1) or (0,0,3).
- If N ≡ 32 (mod 64), then the (sum of exponents of prime factors of N of the form 5a² + 4ab + 31b², sum of exponents of prime factors of N of the form 11a² + 10ab + 16b², sum of exponents of prime factors of N of the form 8a² + 2ab + 19b²) is not (0,0,0) or (0,1,0) or (0,0,2).
- If N ≡ 64 (mod 128), then the (sum of exponents of prime factors of N of the form 5a² + 4ab + 31b², sum of exponents of prime factors of N of the form 11a² + 10ab + 16b², sum of exponents of prime factors of N of the form 8a² + 2ab + 19b²) is not (0,0,1).
- If N ≡ 128 (mod 256), then the (sum of exponents of prime factors of N of the form 5a² + 4ab + 31b², sum of exponents of prime factors of N of the form 11a² + 10ab + 16b², sum of exponents of prime factors of N of the form 8a² + 2ab + 19b²) is not (0,0,0).

k = 157

Primes p of the form a² + 157b²: Some primes congruent to {1, 9, 13, 17, 25, 33, 37, 49, 57, 81, 89, 93, 101, 105, 109, 113, 117, 121, 141, 145, 153, 157, 161, 169, 173, 193, 197, 201, 205, 209, 213, 221, 225, 233, 257, 265, 277, 281, 289, 297, 301, 305, 313, 317, 325, 333, 341, 345, 349, 353, 361, 365, 381, 385, 389, 413, 425, 429, 441, 457, 461, 481, 485, 501, 513, 517, 529, 553, 557, 561, 577, 581, 589, 593, 597, 601, 609, 617, 625} mod 628, not of the form 13a² + 10ab + 14b²
if N is a non-negative integer that can be written as a²+157b², then p × N can be written as a²+157b²
if N is a non-negative integer that cannot be written as a²+157b², then p × N cannot be written as a²+157b²
Primes of the form 13a² + 10ab + 14b²: Other remaining primes congruent to {1, 9, 13, 17, 25, 33, 37, 49, 57, 81, 89, 93, 101, 105, 109, 113, 117, 121, 141, 145, 153, 161, 169, 173, 193, 197, 201, 205, 209, 213, 221, 225, 233, 257, 265, 277, 281, 289, 297, 301, 305, 313, 317, 325, 333, 341, 345, 349, 353, 361, 365, 381, 385, 389, 413, 425, 429, 441, 457, 461, 481, 485, 501, 513, 517, 529, 553, 557, 561, 577, 581, 589, 593, 597, 601, 609, 617, 625} mod 628, not of the form a²+157b²
Primes of the form 7a² + 4ab + 23b²: Some primes congruent to {7, 15, 23, 43, 55, 59, 63, 79, 83, 87, 91, 95, 103, 107, 119, 123, 131, 135, 139, 151, 155, 159, 163, 175, 179, 183, 191, 195, 207, 211, 219, 223, 227, 231, 235, 251, 255, 259, 271, 291, 299, 307, 319, 335, 343, 355, 359, 367, 375, 379, 383, 387, 391, 399, 411, 439, 443, 447, 451, 463, 479, 491, 495, 499, 503, 531, 543, 551, 555, 559, 563, 567, 575, 583, 587, 599, 607, 623} mod 628, not of the form 2a² + 2ab + 79b²
Primes of the form 2a² + 2ab + 79b²: Other remaining primes congruent to {2, 7, 15, 23, 43, 55, 59, 63, 79, 83, 87, 91, 95, 103, 107, 119, 123, 131, 135, 139, 151, 155, 159, 163, 175, 179, 183, 191, 195, 207, 211, 219, 223, 227, 231, 235, 251, 255, 259, 271, 291, 299, 307, 319, 335, 343, 355, 359, 367, 375, 379, 383, 387, 391, 399, 411, 439, 443, 447, 451, 463, 479, 491, 495, 499, 503, 531, 543, 551, 555, 559, 563, 567, 575, 583, 587, 599, 607, 623} mod 628, not of the form 7a² + 4ab + 23b²

N can be written as a² + 157b² if and only if
- N has no prime factors congruent to {3, 5, 11, 19, 21, 27, 29, 31, 35, 39, 41, 45, 47, 51, 53, 61, 65, 67, 69, 71, 73, 75, 77, 85, 97, 99, 111, 115, 125, 127, 129, 133, 137, 143, 147, 149, 165, 167, 171, 177, 181, 185, 187, 189, 199, 203, 215, 217, 229, 237, 239, 241, 243, 245, 247, 249, 253, 261, 263, 267, 269, 273, 275, 279, 283, 285, 287, 293, 295, 303, 309, 311, 315, 321, 323, 327, 329, 331, 337, 339, 347, 351, 357, 363, 369, 371, 373, 377, 393, 395, 397, 401, 403, 405, 407, 409, 415, 417, 419, 421, 423, 427, 431, 433, 435, 437, 445, 449, 453, 455, 459, 465, 467, 469, 473, 475, 477, 483, 487, 489, 493, 497, 505, 507, 509, 511, 515, 519, 521, 523, 525, 527, 533, 535, 537, 539, 541, 545, 547, 549, 565, 569, 571, 573, 579, 585, 591, 595, 603, 605, 611, 613, 615, 619, 621, 627} mod 628 to an odd power.
- Sum of exponents of prime factors of N of the form 7a² + 4ab + 23b² and the sum of exponents of prime factors of N of the form 2a² + 2ab + 79b² are of the same parity.
- Sum of exponents of prime factors of N of the form 7a² + 4ab + 23b² and the sum of exponents of prime factors of N of the form 13a² + 10ab + 14b² do not add up to one.

Raman · 2013-03-23, 23:36

k = 162

Primes p of the form a² + 162b²: Some primes congruent to {1, 19} mod 24, not of the form 9a² + 6ab + 19b²
if N is a non-negative integer that can be written as a²+162b², then p × N can be written as a²+162b²
if N is a non-negative integer that cannot be written as a²+162b², then p × N cannot be written as a²+162b²
Primes of the form 9a² + 6ab + 19b²: Other remaining primes congruent to {1, 19} mod 24, not of the form a²+162b²
Primes of the form 11a² + 10ab + 17b²: Some primes congruent to {11, 17} mod 24, not of the form 2a² + 81b²
Primes of the form 2a² + 81b²: Other remaining primes congruent to {2, 11, 17} mod 24, not of the form 11a² + 10ab + 17b²

N can be written as a² + 162b² if and only if
- N is not congruent to {3, 6} (mod 9) or {27, 54} (mod 81).
- N has no prime factors congruent to {5, 7, 13, 23} mod 24 to an odd power.
- If N is not a multiple of 3 or if N ≡ {9, 18} (mod 27), then the sum of exponents of prime factors of N of the form 11a² + 10ab + 17b² and the sum of exponents of prime factors of N of the form 2a² + 81b² are of the same parity.
- If N is not a multiple of 3, then the sum of exponents of prime factors of N of the form 11a² + 10ab + 17b² and the sum of exponents of prime factors of N of the form 9a² + 6ab + 19b² do not add up to one.

k = 169

Primes p of the form a² + 169b²: Some primes congruent to {1, 9, 17, 25, 29, 49} mod 52, not of the form 10a² + 2ab + 17b²
if N is a non-negative integer that can be written as a²+169b², then p × N can be written as a²+169b²
if N is a non-negative integer that cannot be written as a²+169b², then p × N cannot be written as a²+169b²
Primes of the form 10a² + 2ab + 17b²: Other remaining primes congruent to {1, 9, 17, 25, 29, 49} mod 52, not of the form a²+169b²
Primes of the form 5a² + 2ab + 34b²: Some primes congruent to {5, 21, 33, 37, 41, 45} mod 52, not of the form 2a² + 2ab + 85b²
Primes of the form 2a² + 2ab + 85b²: Other remaining primes congruent to {2, 5, 21, 33, 37, 41, 45} mod 52, not of the form 5a² + 2ab + 34b²

N can be written as a² + 169b² if and only if
- N is not congruent to {13, 26, 39, 52, 65, 78, 91, 104, 117, 130, 143, 156} (mod 169).
- N has no prime factors congruent to {3, 7, 11, 15, 19, 23, 27, 31, 35, 43, 47, 51} mod 52 to an odd power.
- If N is not a multiple of 13, then the sum of exponents of prime factors of N of the form 5a² + 2ab + 34b² and the sum of exponents of prime factors of N of the form 2a² + 2ab + 85b² are of the same parity.
- If N is not a multiple of 13, then the sum of exponents of prime factors of N of the form 5a² + 2ab + 34b² and the sum of exponents of prime factors of N of the form 10a² + 2ab + 17b² do not add up to one.

k = 172

Primes p of the form a² + 172b²: Some primes congruent to {1, 9, 13, 17, 21, 25, 41, 49, 53, 57, 81, 97, 101, 109, 117, 121, 133, 145, 153, 165, 169} mod 172, not of the form 13a² + 12ab + 16b²
if N is a non-negative integer that can be written as a²+172b², then p × N can be written as a²+172b²
if N is a non-negative integer that cannot be written as a²+172b², then p × N cannot be written as a²+172b²
Primes of the form 13a² + 12ab + 16b²: Other remaining primes congruent to {1, 9, 13, 17, 21, 25, 41, 49, 53, 57, 81, 97, 101, 109, 117, 121, 133, 145, 153, 165, 169} mod 172, not of the form a²+172b²
Primes of the form 11a² + 4ab + 16b²: Some primes congruent to {11, 15, 23, 31, 35, 47, 59, 67, 79, 83, 87, 95, 99, 103, 107, 111, 127, 135, 139, 143, 167} mod 172, not of the form 4a² + 43b²
Primes of the form 4a² + 43b²: Other remaining primes congruent to {11, 15, 23, 31, 35, 43, 47, 59, 67, 79, 83, 87, 95, 99, 103, 107, 111, 127, 135, 139, 143, 167} mod 172, not of the form 11a² + 4ab + 16b²

N can be written as a² + 172b² if and only if
- N has no prime factors congruent to {2, 3, 5, 7, 19, 27, 29, 33, 37, 39, 45, 51, 55, 61, 63, 65, 69, 71, 73, 75, 77, 85, 89, 91, 93, 105, 113, 115, 119, 123, 125, 131, 137, 141, 147, 149, 151, 155, 157, 159, 161, 163, 171} mod 172 to an odd power.
- If N is odd, then the sum of exponents of prime factors of N of the form 11a² + 4ab + 16b² and the sum of exponents of prime factors of N of the form 4a² + 43b² are of the same parity.
- If N is odd or N ≡ 4 (mod 8), then the sum of exponents of prime factors of N of the form 11a² + 4ab + 16b² and the sum of exponents of prime factors of N of the form 13a² + 12ab + 16b² do not add up to one.

Raman · 2013-03-23, 23:37

k = 175

Primes p of the form a² + 175b²: Some primes congruent to {1, 9, 11, 29, 39, 51} mod 70, not of the form 11a² + 2ab + 16b²
if N is a non-negative integer that can be written as a²+175b², then p × N can be written as a²+175b²
if N is a non-negative integer that cannot be written as a²+175b², then p × N cannot be written as a²+175b²
Primes of the form 11a² + 2ab + 16b²: Other remaining primes congruent to {1, 9, 11, 29, 39, 51} mod 70, not of the form a²+175b²
Primes of the form 8a² + 6ab + 23b²: Some primes congruent to {23, 37, 43, 53, 57, 67} mod 70, not of the form 7a² + 25b²
Primes of the form 7a² + 25b²: Other remaining primes congruent to {7, 23, 37, 43, 53, 57, 67} mod 70, not of the form 8a² + 6ab + 23b²

N can be written as a² + 175b² if and only if
- N is not congruent to 2 (mod 4).
- N has no prime factors congruent to {3, 5, 13, 17, 19, 27, 31, 33, 41, 47, 59, 61, 69} mod 70 to an odd power.
- If N is not a multiple of 5, and the highest power of 2 dividing N is an even number (or zero), then the sum of exponents of prime factors of N of the form 8a² + 6ab + 23b² and the sum of exponents of prime factors of N of the form 7a² + 25b² are of the same parity.
- If N is not a multiple of 5, and the highest power of 2 dividing N is an odd number that is three or more, then the sum of exponents of prime factors of N of the form 8a² + 6ab + 23b² and the sum of exponents of prime factors of N of the form 7a² + 25b² are of different parity.
- If N ≡ {1, 3, 7, 9} (mod 10) or if N ≡ {4, 12, 28, 36} (mod 40), then the sum of exponents of prime factors of N of the form 8a² + 6ab + 23b² and the sum of exponents of prime factors of N of the form 11a² + 2ab + 16b² do not add up to one.
- If N ≡ {8, 24, 56, 72} (mod 80), then there is atleast one prime factor of N of the form 8a² + 6ab + 23b² or 11a² + 2ab + 16b².

k = 187

Primes p of the form a² + 187b²: Some primes congruent to {1, 4, 9, 15, 16, 25, 26, 36, 38, 42, 47, 49, 53, 59, 60, 64, 67, 69, 70, 81, 86, 89, 93, 100, 103, 104, 111, 115, 135, 137, 144, 152, 155, 157, 166, 168, 169, 174, 179, 185} mod 187, not of the form 4a² + 2ab + 47b²
if N is a non-negative integer that can be written as a²+187b², then p × N can be written as a²+187b²
if N is a non-negative integer that cannot be written as a²+187b², then p × N cannot be written as a²+187b²
Primes of the form 4a² + 2ab + 47b²: Other remaining primes congruent to {1, 4, 9, 15, 16, 25, 26, 36, 38, 42, 47, 49, 53, 59, 60, 64, 67, 69, 70, 81, 86, 89, 93, 100, 103, 104, 111, 115, 135, 137, 144, 152, 155, 157, 166, 168, 169, 174, 179, 185} mod 187, not of the form a²+187b²
Primes of the form 7a² + 6ab + 28b²: Some primes congruent to {6, 7, 10, 24, 28, 29, 39, 40, 41, 46, 54, 57, 61, 62, 63, 65, 73, 74, 79, 90, 95, 96, 105, 107, 109, 112, 116, 129, 131, 139, 142, 150, 156, 160, 164, 167, 173, 175, 182, 184} mod 187, not of the form 11a² + 17b²
Primes of the form 11a² + 17b²: Other remaining primes congruent to {6, 7, 10, 11, 17, 24, 28, 29, 39, 40, 41, 46, 54, 57, 61, 62, 63, 65, 73, 74, 79, 90, 95, 96, 105, 107, 109, 112, 116, 129, 131, 139, 142, 150, 156, 160, 164, 167, 173, 175, 182, 184} mod 187, not of the form 7a² + 6ab + 28b²

N can be written as a² + 187b² if and only if
- N has no prime factors congruent to {2, 3, 5, 8, 12, 13, 14, 18, 19, 20, 21, 23, 27, 30, 31, 32, 35, 37, 43, 45, 48, 50, 52, 56, 58, 71, 72, 75, 76, 78, 80, 82, 83, 84, 87, 91, 92, 94, 97, 98, 101, 106, 108, 113, 114, 117, 118, 120, 122, 123, 124, 125, 126, 127, 128, 130, 133, 134, 138, 140, 141, 145, 146, 147, 148, 149, 151, 158, 159, 161, 162, 163, 171, 172, 177, 178, 180, 181, 183, 186} mod 187 to an odd power.
- Sum of exponents of prime factors of N of the form 7a² + 6ab + 28b² and the sum of exponents of prime factors of N of the form 11a² + 17b² are of the same parity.
- If N is odd, then the sum of exponents of prime factors of N of the form 7a² + 6ab + 28b² and the sum of exponents of prime factors of N of the form 4a² + 2ab + 47b² do not add up to one.

k = 193

Primes p of the form a² + 193b²: Some primes congruent to {1, 9, 21, 25, 49, 65, 69, 81, 85, 93, 97, 101, 109, 121, 129, 137, 145, 157, 161, 165, 169, 177, 181, 185, 189, 193, 197, 201, 205, 209, 217, 221, 225, 229, 241, 249, 257, 265, 277, 285, 289, 293, 301, 305, 317, 321, 337, 361, 365, 377, 385, 389, 393, 409, 413, 417, 429, 441, 445, 449, 453, 461, 469, 481, 493, 517, 525, 529, 533, 537, 561, 565, 573, 577, 581, 585, 593, 597, 621, 625, 629, 633, 641, 665, 677, 689, 697, 705, 709, 713, 717, 729, 741, 745, 749, 765, 769} mod 772, not of the form 2a² + 2ab + 97b²
if N is a non-negative integer that can be written as a²+193b², then p × N can be written as a²+193b²
if N is a non-negative integer that cannot be written as a²+193b², then p × N cannot be written as a²+193b²
Primes of the form 2a² + 2ab + 97b²: Other remaining primes congruent to {1, 2, 9, 21, 25, 49, 65, 69, 81, 85, 93, 97, 101, 109, 121, 129, 137, 145, 157, 161, 165, 169, 177, 181, 185, 189, 197, 201, 205, 209, 217, 221, 225, 229, 241, 249, 257, 265, 277, 285, 289, 293, 301, 305, 317, 321, 337, 361, 365, 377, 385, 389, 393, 409, 413, 417, 429, 441, 445, 449, 453, 461, 469, 481, 493, 517, 525, 529, 533, 537, 561, 565, 573, 577, 581, 585, 593, 597, 621, 625, 629, 633, 641, 665, 677, 689, 697, 705, 709, 713, 717, 729, 741, 745, 749, 765, 769} mod 772, not of the form a²+193b²
Primes of the form 11a² + 8ab + 19b²: All primes congruent to {11, 15, 19, 35, 39, 47, 51, 71, 79, 87, 91, 99, 103, 111, 115, 119, 123, 127, 135, 155, 159, 163, 167, 171, 183, 203, 215, 219, 223, 227, 231, 251, 259, 263, 267, 271, 275, 283, 287, 295, 299, 307, 315, 335, 339, 347, 351, 367, 371, 375, 391, 399, 403, 415, 419, 423, 427, 431, 439, 443, 447, 459, 463, 475, 491, 499, 503, 511, 519, 527, 535, 539, 559, 599, 619, 623, 631, 639, 647, 655, 659, 667, 683, 695, 699, 711, 715, 719, 727, 731, 735, 739, 743, 755, 759, 767} mod 772

N can be written as a² + 193b² if and only if
- N has no prime factors congruent to {3, 5, 7, 13, 17, 23, 27, 29, 31, 33, 37, 41, 43, 45, 53, 55, 57, 59, 61, 63, 67, 73, 75, 77, 83, 89, 95, 105, 107, 113, 117, 125, 131, 133, 139, 141, 143, 147, 149, 151, 153, 173, 175, 179, 187, 191, 195, 199, 207, 211, 213, 233, 235, 237, 239, 243, 245, 247, 253, 255, 261, 269, 273, 279, 281, 291, 297, 303, 309, 311, 313, 319, 323, 325, 327, 329, 331, 333, 341, 343, 345, 349, 353, 355, 357, 359, 363, 369, 373, 379, 381, 383, 387, 395, 397, 401, 405, 407, 411, 421, 425, 433, 435, 437, 451, 455, 457, 465, 467, 471, 473, 477, 479, 483, 485, 487, 489, 495, 497, 501, 505, 507, 509, 513, 515, 521, 523, 531, 541, 543, 545, 547, 549, 551, 553, 555, 557, 563, 567, 569, 571, 575, 583, 587, 589, 591, 595, 601, 603, 605, 607, 609, 611, 613, 615, 617, 627, 635, 637, 643, 645, 649, 651, 653, 657, 661, 663, 669, 671, 673, 675, 679, 681, 685, 687, 691, 693, 701, 703, 707, 721, 723, 725, 733, 737, 747, 751, 753, 757, 761, 763, 771} mod 772 to an odd power.
- Sum of exponents of {11, 15, 19, 35, 39, 47, 51, 71, 79, 87, 91, 99, 103, 111, 115, 119, 123, 127, 135, 155, 159, 163, 167, 171, 183, 203, 215, 219, 223, 227, 231, 251, 259, 263, 267, 271, 275, 283, 287, 295, 299, 307, 315, 335, 339, 347, 351, 367, 371, 375, 391, 399, 403, 415, 419, 423, 427, 431, 439, 443, 447, 459, 463, 475, 491, 499, 503, 511, 519, 527, 535, 539, 559, 599, 619, 623, 631, 639, 647, 655, 659, 667, 683, 695, 699, 711, 715, 719, 727, 731, 735, 739, 743, 755, 759, 767} mod 772 prime factors of N is even.
- If N has no prime factors congruent to {11, 15, 19, 35, 39, 47, 51, 71, 79, 87, 91, 99, 103, 111, 115, 119, 123, 127, 135, 155, 159, 163, 167, 171, 183, 203, 215, 219, 223, 227, 231, 251, 259, 263, 267, 271, 275, 283, 287, 295, 299, 307, 315, 335, 339, 347, 351, 367, 371, 375, 391, 399, 403, 415, 419, 423, 427, 431, 439, 443, 447, 459, 463, 475, 491, 499, 503, 511, 519, 527, 535, 539, 559, 599, 619, 623, 631, 639, 647, 655, 659, 667, 683, 695, 699, 711, 715, 719, 727, 731, 735, 739, 743, 755, 759, 767} mod 772, then sum of exponents of prime factors of N of the form 2a² + 2ab + 97b² is even.

2013-03-21, 19:28	#112
Raman Noodles "Mr. Tuch" Dec 2007 Chennai, India 3×419 Posts	128 - 9², 128 - 7², 128 - 5², 128 - 1² k = 47 Primes p of the form a² + 47b²: Some primes congruent to {1, 3, 7, 9, 17, 21, 25, 27, 37, 47, 49, 51, 53, 55, 59, 61, 63, 65, 71, 75, 79, 81, 83, 89} mod 94, neither of the form 3a² + 2ab + 16b² nor 7a² + 6ab + 8b² if N is a non-negative integer that can be written as a²+47b², then p × N can be written as a²+47b² if N is a non-negative integer that cannot be written as a²+46b², then p × N cannot be written as a²+47b² Primes of the form 3a² + 2ab + 16b²: Some more primes congruent to {1, 3, 7, 9, 17, 21, 25, 27, 37, 49, 51, 53, 55, 59, 61, 63, 65, 71, 75, 79, 81, 83, 89} mod 94, neither of the form a²+47b² nor 7a² + 6ab + 8b² Primes of the form 7a² + 6ab + 8b²: Other remaining primes congruent to {1, 3, 7, 9, 17, 21, 25, 27, 37, 49, 51, 53, 55, 59, 61, 63, 65, 71, 75, 79, 81, 83, 89} mod 94, neither of the form a²+47b² nor 3a² + 2ab + 16b² N can be written as a² + 47b² if and only if - N is not congruent to 2 (mod 4). - N has no prime factors congruent to {5, 11, 13, 15, 19, 23, 29, 31, 33, 35, 39, 41, 43, 45, 57, 67, 69, 73, 77, 85, 87, 91, 93} mod 94 to an odd power. - If N is odd or N ≡ 4 (mod 8), then the (sum of exponents of prime factors of N of the form 3a² + 2ab + 16b², sum of exponents of prime factors of N of the form 7a² + 6ab + 8b²) is not (0,1) or (1,0) or (1,1) or (3,0) or (0,3). - If N ≡ 8 (mod 16), then the (sum of exponents of prime factors of N of the form 3a² + 2ab + 16b², sum of exponents of prime factors of N of the form 7a² + 6ab + 8b²) is not (0,0) or (1,0) or (0,2). - If N ≡ 16 (mod 32), then the (sum of exponents of prime factors of N of the form 3a² + 2ab + 16b², sum of exponents of prime factors of N of the form 7a² + 6ab + 8b²) is not (0,1). - If N ≡ 32 (mod 64), then the (sum of exponents of prime factors of N of the form 3a² + 2ab + 16b², sum of exponents of prime factors of N of the form 7a² + 6ab + 8b²) is not (0,0). k = 79 Primes p of the form a² + 79b²: Some primes congruent to {1, 5, 9, 11, 13, 19, 21, 23, 25, 31, 45, 49, 51, 55, 65, 67, 73, 79, 81, 83, 87, 89, 95, 97, 99, 101, 105, 111, 115, 117, 119, 121, 123, 125, 129, 131, 141, 143, 151, 155} mod 158, neither of the form 5a² + 2ab + 16b² nor 8a² + 6ab + 11b² if N is a non-negative integer that can be written as a²+79b², then p × N can be written as a²+79b² if N is a non-negative integer that cannot be written as a²+79b², then p × N cannot be written as a²+79b² Primes of the form 5a² + 2ab + 16b²: Some more primes congruent to {1, 5, 9, 11, 13, 19, 21, 23, 25, 31, 45, 49, 51, 55, 65, 67, 73, 81, 83, 87, 89, 95, 97, 99, 101, 105, 111, 115, 117, 119, 121, 123, 125, 129, 131, 141, 143, 151, 155} mod 158, neither of the form a²+79b² nor 8a² + 6ab + 11b² Primes of the form 8a² + 6ab + 11b²: Other remaining primes congruent to {1, 5, 9, 11, 13, 19, 21, 23, 25, 31, 45, 49, 51, 55, 65, 67, 73, 81, 83, 87, 89, 95, 97, 99, 101, 105, 111, 115, 117, 119, 121, 123, 125, 129, 131, 141, 143, 151, 155} mod 158, neither of the form a²+79b² nor 5a² + 2ab + 16b² N can be written as a² + 79b² if and only if - N is not congruent to 2 (mod 4). - N has no prime factors congruent to {3, 7, 15, 17, 27, 29, 33, 35, 37, 39, 41, 43, 47, 53, 57, 59, 61, 63, 69, 71, 75, 77, 85, 91, 93, 103, 107, 109, 113, 127, 133, 135, 137, 139, 145, 147, 149, 153, 157} mod 158 to an odd power. - If N is odd or N ≡ 4 (mod 8), then the (sum of exponents of prime factors of N of the form 5a² + 2ab + 16b², sum of exponents of prime factors of N of the form 8a² + 6ab + 11b²) is not (0,1) or (1,0) or (1,1) or (3,0) or (0,3). - If N ≡ 8 (mod 16), then the (sum of exponents of prime factors of N of the form 5a² + 2ab + 16b², sum of exponents of prime factors of N of the form 8a² + 6ab + 11b²) is not (0,0) or (1,0) or (0,2). - If N ≡ 16 (mod 32), then the (sum of exponents of prime factors of N of the form 5a² + 2ab + 16b², sum of exponents of prime factors of N of the form 8a² + 6ab + 11b²) is not (0,1). - If N ≡ 32 (mod 64), then the (sum of exponents of prime factors of N of the form 5a² + 2ab + 16b², sum of exponents of prime factors of N of the form 8a² + 6ab + 11b²) is not (0,0). k = 103 Primes p of the form a² + 103b²: Some primes congruent to {1, 7, 9, 13, 15, 17, 19, 23, 25, 29, 33, 41, 49, 55, 59, 61, 63, 79, 81, 83, 91, 93, 97, 103, 105, 107, 111, 117, 119, 121, 129, 131, 133, 135, 137, 139, 141, 149, 153, 155, 159, 161, 163, 167, 169, 171, 175, 179, 185, 195, 201, 203} mod 206, neither of the form 7a² + 6ab + 16b² nor 8a² + 2ab + 13b² if N is a non-negative integer that can be written as a²+103b², then p × N can be written as a²+103b² if N is a non-negative integer that cannot be written as a²+103b², then p × N cannot be written as a²+103b² Primes of the form 7a² + 6ab + 16b²: Some more primes congruent to {1, 7, 9, 13, 15, 17, 19, 23, 25, 29, 33, 41, 49, 55, 59, 61, 63, 79, 81, 83, 91, 93, 97, 105, 107, 111, 117, 119, 121, 129, 131, 133, 135, 137, 139, 141, 149, 153, 155, 159, 161, 163, 167, 169, 171, 175, 179, 185, 195, 201, 203} mod 206, neither of the form a²+103b² nor 8a² + 2ab + 13b² Primes of the form 8a² + 2ab + 13b²: Other remaining primes congruent to {1, 7, 9, 13, 15, 17, 19, 23, 25, 29, 33, 41, 49, 55, 59, 61, 63, 79, 81, 83, 91, 93, 97, 105, 107, 111, 117, 119, 121, 129, 131, 133, 135, 137, 139, 141, 149, 153, 155, 159, 161, 163, 167, 169, 171, 175, 179, 185, 195, 201, 203} mod 206, neither of the form a²+103b² nor 7a² + 6ab + 16b² N can be written as a² + 103b² if and only if - N is not congruent to 2 (mod 4). - N has no prime factors congruent to {3, 5, 11, 21, 27, 31, 35, 37, 39, 43, 45, 47, 51, 53, 57, 65, 67, 69, 71, 73, 75, 77, 85, 87, 89, 95, 99, 101, 109, 113, 115, 123, 125, 127, 143, 145, 147, 151, 157, 165, 173, 177, 181, 183, 187, 189, 191, 193, 197, 199, 205} mod 206 to an odd power. - If N is odd or N ≡ 4 (mod 8), then the (sum of exponents of prime factors of N of the form 7a² + 6ab + 16b², sum of exponents of prime factors of N of the form 8a² + 2ab + 13b²) is not (0,1) or (1,0) or (1,1) or (3,0) or (0,3). - If N ≡ 8 (mod 16), then the (sum of exponents of prime factors of N of the form 7a² + 6ab + 16b², sum of exponents of prime factors of N of the form 8a² + 2ab + 13b²) is not (0,0) or (1,0) or (0,2). - If N ≡ 16 (mod 32), then the (sum of exponents of prime factors of N of the form 7a² + 6ab + 16b², sum of exponents of prime factors of N of the form 8a² + 2ab + 13b²) is not (0,1). - If N ≡ 32 (mod 64), then the (sum of exponents of prime factors of N of the form 7a² + 6ab + 16b², sum of exponents of prime factors of N of the form 8a² + 2ab + 13b²) is not (0,0). k = 127 Primes p of the form a² + 127b²: Some primes congruent to {1, 9, 11, 13, 15, 17, 19, 21, 25, 31, 35, 37, 41, 47, 49, 61, 69, 71, 73, 79, 81, 87, 99, 103, 107, 113, 115, 117, 121, 127, 129, 131, 135, 143, 145, 149, 153, 157, 159, 161, 163, 165, 169, 171, 177, 179, 187, 189, 191, 195, 197, 199, 201, 203, 209, 211, 215, 221, 225, 227, 231, 247, 249, 251} mod 254, neither of the form 11a² + 8ab + 13b² nor 8a² + 6ab + 17b² if N is a non-negative integer that can be written as a²+127b², then p × N can be written as a²+127b² if N is a non-negative integer that cannot be written as a²+127b², then p × N cannot be written as a²+127b² Primes of the form 11a² + 8ab + 13b²: Some more primes congruent to {1, 9, 11, 13, 15, 17, 19, 21, 25, 31, 35, 37, 41, 47, 49, 61, 69, 71, 73, 79, 81, 87, 99, 103, 107, 113, 115, 117, 121, 129, 131, 135, 143, 145, 149, 153, 157, 159, 161, 163, 165, 169, 171, 177, 179, 187, 189, 191, 195, 197, 199, 201, 203, 209, 211, 215, 221, 225, 227, 231, 247, 249, 251} mod 254, neither of the form a²+127b² nor 8a² + 6ab + 17b² Primes of the form 8a² + 6ab + 17b²: Other remaining primes congruent to {1, 9, 11, 13, 15, 17, 19, 21, 25, 31, 35, 37, 41, 47, 49, 61, 69, 71, 73, 79, 81, 87, 99, 103, 107, 113, 115, 117, 121, 129, 131, 135, 143, 145, 149, 153, 157, 159, 161, 163, 165, 169, 171, 177, 179, 187, 189, 191, 195, 197, 199, 201, 203, 209, 211, 215, 221, 225, 227, 231, 247, 249, 251} mod 254, neither of the form a²+127b² nor 11a² + 8ab + 13b² N can be written as a² + 127b² if and only if - N is not congruent to 2 (mod 4). - N has no prime factors congruent to {3, 5, 7, 23, 27, 29, 33, 39, 43, 45, 51, 53, 55, 57, 59, 63, 65, 67, 75, 77, 83, 85, 89, 91, 93, 95, 97, 101, 105, 109, 111, 119, 123, 125, 133, 137, 139, 141, 147, 151, 155, 167, 173, 175, 181, 183, 185, 193, 205, 207, 213, 217, 219, 223, 229, 233, 235, 237, 239, 241, 243, 245, 253} mod 254 to an odd power. - If N is odd or N ≡ 4 (mod 8), then the (sum of exponents of prime factors of N of the form 11a² + 8ab + 13b², sum of exponents of prime factors of N of the form 8a² + 6ab + 17b²) is not (0,1) or (1,0) or (1,1) or (3,0) or (0,3). - If N ≡ 8 (mod 16), then the (sum of exponents of prime factors of N of the form 11a² + 8ab + 13b², sum of exponents of prime factors of N of the form 8a² + 6ab + 17b²) is not (0,0) or (1,0) or (0,2). - If N ≡ 16 (mod 32), then the (sum of exponents of prime factors of N of the form 11a² + 8ab + 13b², sum of exponents of prime factors of N of the form 8a² + 6ab + 17b²) is not (0,1). - If N ≡ 32 (mod 64), then the (sum of exponents of prime factors of N of the form 11a² + 8ab + 13b², sum of exponents of prime factors of N of the form 8a² + 6ab + 17b²) is not (0,0).

2013-03-23, 23:36	#120
Raman Noodles "Mr. Tuch" Dec 2007 Chennai, India 3·419 Posts	k = 162 Primes p of the form a² + 162b²: Some primes congruent to {1, 19} mod 24, not of the form 9a² + 6ab + 19b² if N is a non-negative integer that can be written as a²+162b², then p × N can be written as a²+162b² if N is a non-negative integer that cannot be written as a²+162b², then p × N cannot be written as a²+162b² Primes of the form 9a² + 6ab + 19b²: Other remaining primes congruent to {1, 19} mod 24, not of the form a²+162b² Primes of the form 11a² + 10ab + 17b²: Some primes congruent to {11, 17} mod 24, not of the form 2a² + 81b² Primes of the form 2a² + 81b²: Other remaining primes congruent to {2, 11, 17} mod 24, not of the form 11a² + 10ab + 17b² N can be written as a² + 162b² if and only if - N is not congruent to {3, 6} (mod 9) or {27, 54} (mod 81). - N has no prime factors congruent to {5, 7, 13, 23} mod 24 to an odd power. - If N is not a multiple of 3 or if N ≡ {9, 18} (mod 27), then the sum of exponents of prime factors of N of the form 11a² + 10ab + 17b² and the sum of exponents of prime factors of N of the form 2a² + 81b² are of the same parity. - If N is not a multiple of 3, then the sum of exponents of prime factors of N of the form 11a² + 10ab + 17b² and the sum of exponents of prime factors of N of the form 9a² + 6ab + 19b² do not add up to one. k = 169 Primes p of the form a² + 169b²: Some primes congruent to {1, 9, 17, 25, 29, 49} mod 52, not of the form 10a² + 2ab + 17b² if N is a non-negative integer that can be written as a²+169b², then p × N can be written as a²+169b² if N is a non-negative integer that cannot be written as a²+169b², then p × N cannot be written as a²+169b² Primes of the form 10a² + 2ab + 17b²: Other remaining primes congruent to {1, 9, 17, 25, 29, 49} mod 52, not of the form a²+169b² Primes of the form 5a² + 2ab + 34b²: Some primes congruent to {5, 21, 33, 37, 41, 45} mod 52, not of the form 2a² + 2ab + 85b² Primes of the form 2a² + 2ab + 85b²: Other remaining primes congruent to {2, 5, 21, 33, 37, 41, 45} mod 52, not of the form 5a² + 2ab + 34b² N can be written as a² + 169b² if and only if - N is not congruent to {13, 26, 39, 52, 65, 78, 91, 104, 117, 130, 143, 156} (mod 169). - N has no prime factors congruent to {3, 7, 11, 15, 19, 23, 27, 31, 35, 43, 47, 51} mod 52 to an odd power. - If N is not a multiple of 13, then the sum of exponents of prime factors of N of the form 5a² + 2ab + 34b² and the sum of exponents of prime factors of N of the form 2a² + 2ab + 85b² are of the same parity. - If N is not a multiple of 13, then the sum of exponents of prime factors of N of the form 5a² + 2ab + 34b² and the sum of exponents of prime factors of N of the form 10a² + 2ab + 17b² do not add up to one. k = 172 Primes p of the form a² + 172b²: Some primes congruent to {1, 9, 13, 17, 21, 25, 41, 49, 53, 57, 81, 97, 101, 109, 117, 121, 133, 145, 153, 165, 169} mod 172, not of the form 13a² + 12ab + 16b² if N is a non-negative integer that can be written as a²+172b², then p × N can be written as a²+172b² if N is a non-negative integer that cannot be written as a²+172b², then p × N cannot be written as a²+172b² Primes of the form 13a² + 12ab + 16b²: Other remaining primes congruent to {1, 9, 13, 17, 21, 25, 41, 49, 53, 57, 81, 97, 101, 109, 117, 121, 133, 145, 153, 165, 169} mod 172, not of the form a²+172b² Primes of the form 11a² + 4ab + 16b²: Some primes congruent to {11, 15, 23, 31, 35, 47, 59, 67, 79, 83, 87, 95, 99, 103, 107, 111, 127, 135, 139, 143, 167} mod 172, not of the form 4a² + 43b² Primes of the form 4a² + 43b²: Other remaining primes congruent to {11, 15, 23, 31, 35, 43, 47, 59, 67, 79, 83, 87, 95, 99, 103, 107, 111, 127, 135, 139, 143, 167} mod 172, not of the form 11a² + 4ab + 16b² N can be written as a² + 172b² if and only if - N has no prime factors congruent to {2, 3, 5, 7, 19, 27, 29, 33, 37, 39, 45, 51, 55, 61, 63, 65, 69, 71, 73, 75, 77, 85, 89, 91, 93, 105, 113, 115, 119, 123, 125, 131, 137, 141, 147, 149, 151, 155, 157, 159, 161, 163, 171} mod 172 to an odd power. - If N is odd, then the sum of exponents of prime factors of N of the form 11a² + 4ab + 16b² and the sum of exponents of prime factors of N of the form 4a² + 43b² are of the same parity. - If N is odd or N ≡ 4 (mod 8), then the sum of exponents of prime factors of N of the form 11a² + 4ab + 16b² and the sum of exponents of prime factors of N of the form 13a² + 12ab + 16b² do not add up to one. Last fiddled with by Raman on 2013-03-23 at 23:38

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2013-03-21, 19:29	#113
Raman Noodles "Mr. Tuch" Dec 2007 Chennai, India 4E9₁₆ Posts	k = 71 Primes p of the form a² + 71b²: Some primes congruent to {1, 3, 5, 9, 15, 19, 25, 27, 29, 37, 43, 45, 49, 57, 71, 73, 75, 77, 79, 81, 83, 87, 89, 91, 95, 101, 103, 107, 109, 111, 119, 121, 125, 129, 131, 135} mod 142, neither of the form 3a² + 2ab + 24b² nor 5a² + 4ab + 15b² nor 8a² + 2ab + 9b² if N is a non-negative integer that can be written as a²+71b², then p × N can be written as a²+71b² if N is a non-negative integer that cannot be written as a²+71b², then p × N cannot be written as a²+71b² Primes of the form 3a² + 2ab + 24b²: Some more primes congruent to {1, 3, 5, 9, 15, 19, 25, 27, 29, 37, 43, 45, 49, 57, 73, 75, 77, 79, 81, 83, 87, 89, 91, 95, 101, 103, 107, 109, 111, 119, 121, 125, 129, 131, 135} mod 142, neither of the form a²+71b² nor 5a² + 4ab + 15b² nor 8a² + 2ab + 9b² Primes of the form 5a² + 4ab + 15b²: Some more primes congruent to {1, 3, 5, 9, 15, 19, 25, 27, 29, 37, 43, 45, 49, 57, 73, 75, 77, 79, 81, 83, 87, 89, 91, 95, 101, 103, 107, 109, 111, 119, 121, 125, 129, 131, 135} mod 142, neither of the form a²+71b² nor 3a² + 2ab + 24b² nor 8a² + 2ab + 9b² Primes of the form 8a² + 2ab + 9b²: Other remaining primes congruent to {1, 3, 5, 9, 15, 19, 25, 27, 29, 37, 43, 45, 49, 57, 73, 75, 77, 79, 81, 83, 87, 89, 91, 95, 101, 103, 107, 109, 111, 119, 121, 125, 129, 131, 135} mod 142, neither of the form a²+71b² nor 3a² + 2ab + 24b² nor 5a² + 4ab + 15b² N can be written as a² + 71b² if and only if - N is not congruent to 2 (mod 4). - N has no prime factors congruent to {7, 11, 13, 17, 21, 23, 31, 33, 35, 39, 41, 47, 51, 53, 55, 59, 61, 63, 65, 67, 69, 85, 93, 97, 99, 105, 113, 115, 117, 123, 127, 133, 137, 139, 141} mod 142 to an odd power. - If N is odd or N ≡ 4 (mod 8), then the (sum of exponents of prime factors of N of the form 3a² + 2ab + 24b², sum of exponents of prime factors of N of the form 5a² + 4ab + 15b², sum of exponents of prime factors of N of the form 8a² + 2ab + 9b²) is not (1,0,0) or (0,1,0) or (0,0,1) or (1,1,0) or (1,0,1) or (0,1,1) or (3,0,0) or (0,3,0) or (0,0,3) or (2,1,0) or (0,2,1) or (1,0,2) or (1,3,0) or (0,1,3) or (3,0,1) or (5,0,0) or (0,5,0). - If N ≡ 8 (mod 16), then the (sum of exponents of prime factors of N of the form 3a² + 2ab + 24b², sum of exponents of prime factors of N of the form 5a² + 4ab + 15b², sum of exponents of prime factors of N of the form 8a² + 2ab + 9b²) is not (0,0,0) or (1,0,0) or (0,1,0) or (1,0,1) or (0,2,0) or (0,0,2) or (0,1,2) or (3,0,0). - If N ≡ 16 (mod 32), then the (sum of exponents of prime factors of N of the form 3a² + 2ab + 24b², sum of exponents of prime factors of N of the form 5a² + 4ab + 15b², sum of exponents of prime factors of N of the form 8a² + 2ab + 9b²) is not (1,0,0) or (0,0,1) or (0,1,1) or (0,0,3). - If N ≡ 32 (mod 64), then the (sum of exponents of prime factors of N of the form 3a² + 2ab + 24b², sum of exponents of prime factors of N of the form 5a² + 4ab + 15b², sum of exponents of prime factors of N of the form 8a² + 2ab + 9b²) is not (0,0,0) or (0,1,0) or (0,0,2). - If N ≡ 64 (mod 128), then the (sum of exponents of prime factors of N of the form 3a² + 2ab + 24b², sum of exponents of prime factors of N of the form 5a² + 4ab + 15b², sum of exponents of prime factors of N of the form 8a² + 2ab + 9b²) is not (0,0,1). - If N ≡ 128 (mod 256), then the (sum of exponents of prime factors of N of the form 3a² + 2ab + 24b², sum of exponents of prime factors of N of the form 5a² + 4ab + 15b², sum of exponents of prime factors of N of the form 8a² + 2ab + 9b²) is not (0,0,0). k = 115 Primes p of the form a² + 115b²: Some primes congruent to {1, 4, 6, 9, 16, 24, 26, 29, 31, 36, 39, 41, 49, 54, 59, 64, 71, 81, 94, 96, 101, 104} mod 115, not of the form 4a² + 2ab + 29b² if N is a non-negative integer that can be written as a²+115b², then p × N can be written as a²+115b² if N is a non-negative integer that cannot be written as a²+115b², then p × N cannot be written as a²+115b² Primes of the form 4a² + 2ab + 29b²: Other remaining primes congruent to {1, 4, 6, 9, 16, 24, 26, 29, 31, 36, 39, 41, 49, 54, 59, 64, 71, 81, 94, 96, 101, 104} mod 115, not of the form a²+115b² Primes of the form 7a² + 4ab + 17b²: Some primes congruent to {7, 17, 22, 28, 33, 37, 38, 42, 43, 53, 57, 63, 67, 68, 83, 88, 97, 102, 103, 107, 112, 113} mod 115, not of the form 5a² + 23b² Primes of the form 5a² + 23b²: Other remaining primes congruent to {5, 7, 17, 22, 23, 28, 33, 37, 38, 42, 43, 53, 57, 63, 67, 68, 83, 88, 97, 102, 103, 107, 112, 113} mod 115, not of the form 7a² + 4ab + 17b² N can be written as a² + 115b² if and only if - N has no prime factors congruent to {2, 3, 8, 11, 12, 13, 14, 18, 19, 21, 27, 32, 34, 44, 47, 48, 51, 52, 56, 58, 61, 62, 66, 72, 73, 74, 76, 77, 78, 79, 82, 84, 86, 87, 89, 91, 93, 98, 99, 106, 108, 109, 111, 114} mod 115 to an odd power. - Sum of exponents of prime factors of N of the form 7a² + 4ab + 17b² and the sum of exponents of prime factors of N of the form 5a² + 23b² are of the same parity. - If N is odd, then the sum of exponents of prime factors of N of the form 7a² + 4ab + 17b² and the sum of exponents of prime factors of N of the form 4a² + 2ab + 29b² do not add up to one. k = 123 Primes p of the form a² + 123b²: Some primes congruent to {1, 4, 10, 16, 25, 31, 37, 40, 43, 46, 49, 61, 64, 73, 91, 100, 103, 115, 118, 121} mod 123, not of the form 4a² + 2ab + 31b² if N is a non-negative integer that can be written as a²+123b², then p × N can be written as a²+123b² if N is a non-negative integer that cannot be written as a²+123b², then p × N cannot be written as a²+123b² Primes of the form 4a² + 2ab + 31b²: Other remaining primes congruent to {1, 4, 10, 16, 25, 31, 37, 40, 43, 46, 49, 61, 64, 73, 91, 100, 103, 115, 118, 121} mod 123, not of the form a²+123b² Primes of the form 11a² + 6ab + 12b²: Some primes congruent to {11, 14, 17, 26, 29, 35, 38, 44, 47, 53, 56, 65, 68, 71, 89, 95, 101, 104, 110, 116} mod 123, not of the form 3a² + 41b² Primes of the form 3a² + 41b²: Other remaining primes congruent to {3, 11, 14, 17, 26, 29, 35, 38, 41, 44, 47, 53, 56, 65, 68, 71, 89, 95, 101, 104, 110, 116} mod 123, not of the form 11a² + 6ab + 12b² N can be written as a² + 123b² if and only if - N has no prime factors congruent to {2, 5, 7, 8, 13, 19, 20, 22, 23, 28, 32, 34, 50, 52, 55, 58, 59, 62, 67, 70, 74, 76, 77, 79, 80, 83, 85, 86, 88, 92, 94, 97, 98, 106, 107, 109, 112, 113, 119, 122} mod 123 to an odd power. - Sum of exponents of prime factors of N of the form 11a² + 6ab + 12b² and the sum of exponents of prime factors of N of the form 3a² + 41b² are of the same parity. - If N is odd, then the sum of exponents of prime factors of N of the form 11a² + 6ab + 12b² and the sum of exponents of prime factors of N of the form 4a² + 2ab + 31b² do not add up to one. k = 124 Primes p of the form a² + 124b²: Some primes congruent to {1, 5, 9, 25, 33, 41, 45, 49, 69, 81, 97, 101, 109, 113, 121} mod 124, not of the form 5a² + 2ab + 25b² if N is a non-negative integer that can be written as a²+124b², then p × N can be written as a²+124b² if N is a non-negative integer that cannot be written as a²+124b², then p × N cannot be written as a²+124b² Primes of the form 5a² + 2ab + 25b²: Other remaining primes congruent to {1, 5, 9, 25, 33, 41, 45, 49, 69, 81, 97, 101, 109, 113, 121} mod 124, not of the form a²+124b² Primes of the form 7a² + 6ab + 19b²: Some primes congruent to {7, 19, 35, 39, 47, 51, 59, 63, 67, 71, 87, 95, 103, 107, 111} mod 124, not of the form 4a² + 31b² Primes of the form 4a² + 31b²: Other remaining primes congruent to {7, 19, 31, 35, 39, 47, 51, 59, 63, 67, 71, 87, 95, 103, 107, 111} mod 124, not of the form 7a² + 6ab + 19b² N can be written as a² + 124b² if and only if - N is not congruent to 2 (mod 4) or 8 (mod 16). - N has no prime factors congruent to {3, 11, 13, 15, 17, 21, 23, 27, 29, 37, 43, 53, 55, 57, 61, 65, 73, 75, 77, 79, 83, 85, 89, 91, 99, 105, 115, 117, 119, 123} mod 124 to an odd power. - If N is odd, then the sum of exponents of prime factors of N of the form 7a² + 6ab + 19b² and the sum of exponents of prime factors of N of the form 4a² + 31b² are of the same parity. - If N is odd or N ≡ 4 (mod 8) or N ≡ 16 (mod 32), then the sum of exponents of prime factors of N of the form 7a² + 6ab + 19b² and the sum of exponents of prime factors of N of the form 5a² + 2ab + 25b² do not add up to one. - If N ≡ 32 (mod 64), then there is atleast one prime factor of N of the form 7a² + 6ab + 19b² or 5a² + 2ab + 25b².

2013-03-22, 19:19	#114
Raman Noodles "Mr. Tuch" Dec 2007 Chennai, India 1257₁₀ Posts	k = 106 Primes p of the form a² + 106b²: Some primes congruent to {1, 9, 11, 17, 25, 43, 49, 57, 59, 81, 89, 91, 97, 99, 105, 107, 113, 115, 121, 123, 131, 153, 155, 163, 169, 187, 195, 201, 203, 211, 219, 225, 227, 241, 249, 259, 275, 281, 289, 305, 307, 329, 331, 347, 355, 361, 377, 387, 395, 409, 411, 417} mod 424, not of the form 10a² + 4ab + 11b² if N is a non-negative integer that can be written as a²+106b², then p × N can be written as a²+106b² if N is a non-negative integer that cannot be written as a²+106b², then p × N cannot be written as a²+106b² Primes of the form 10a² + 4ab + 11b²: Other remaining primes congruent to {1, 9, 11, 17, 25, 43, 49, 57, 59, 81, 89, 91, 97, 99, 105, 107, 113, 115, 121, 123, 131, 153, 155, 163, 169, 187, 195, 201, 203, 211, 219, 225, 227, 241, 249, 259, 275, 281, 289, 305, 307, 329, 331, 347, 355, 361, 377, 387, 395, 409, 411, 417} mod 424, not of the form a²+106b² Primes of the form 5a² + 4ab + 22b²: Some primes congruent to {5, 21, 23, 31, 39, 45, 55, 61, 71, 79, 85, 87, 101, 103, 109, 111, 125, 127, 133, 141, 151, 157, 167, 173, 181, 189, 191, 207, 215, 231, 239, 245, 247, 253, 263, 277, 279, 285, 287, 295, 341, 349, 351, 357, 359, 373, 383, 389, 391, 397, 405, 421} mod 424, not of the form 2a² + 53b² Primes of the form 2a² + 53b²: Other remaining primes congruent to {2, 5, 21, 23, 31, 39, 45, 53, 55, 61, 71, 79, 85, 87, 101, 103, 109, 111, 125, 127, 133, 141, 151, 157, 167, 173, 181, 189, 191, 207, 215, 231, 239, 245, 247, 253, 263, 277, 279, 285, 287, 295, 341, 349, 351, 357, 359, 373, 383, 389, 391, 397, 405, 421} mod 424, not of the form 5a² + 4ab + 22b² N can be written as a² + 106b² if and only if - N has no prime factors congruent to {3, 7, 13, 15, 19, 27, 29, 33, 35, 37, 41, 47, 51, 63, 65, 67, 69, 73, 75, 77, 83, 93, 95, 117, 119, 129, 135, 137, 139, 143, 145, 147, 149, 161, 165, 171, 175, 177, 179, 183, 185, 193, 197, 199, 205, 209, 213, 217, 221, 223, 229, 233, 235, 237, 243, 251, 255, 257, 261, 267, 269, 271, 273, 283, 291, 293, 297, 299, 301, 303, 309, 311, 313, 315, 317, 319, 321, 323, 325, 327, 333, 335, 337, 339, 343, 345, 353, 363, 365, 367, 369, 375, 379, 381, 385, 393, 399, 401, 403, 407, 413, 415, 419, 423} mod 424 to an odd power. - Sum of exponents of prime factors of N of the form 5a² + 4ab + 22b² and the sum of exponents of prime factors of N of the form 2a² + 53b² are of the same parity. - Sum of exponents of prime factors of N of the form 5a² + 4ab + 22b² and the sum of exponents of prime factors of N of the form 10a² + 4ab + 11b² do not add up to one. k = 108 Primes p of the form a² + 108b²: Some primes congruent to {1} mod 12, not of the form 9a² + 6ab + 13b² if N is a non-negative integer that can be written as a²+108b², then p × N can be written as a²+108b² if N is a non-negative integer that cannot be written as a²+108b², then p × N cannot be written as a²+108b² Primes of the form 9a² + 6ab + 13b²: Other remaining primes congruent to {1} mod 12, not of the form a²+108b² Primes of the form 7a² + 4ab + 16b²: Some primes congruent to {7} mod 12, not of the form 4a² + 27b² Primes of the form 4a² + 27b²: Other remaining primes congruent to {7} mod 12, not of the form 7a² + 4ab + 16b² N can be written as a² + 108b² if and only if - N is not congruent to {3, 6} mod 9. - N has no prime factors congruent to {2, 5, 11} mod 12 to an odd power. - If N is odd, then the sum of exponents of {3, 7} mod 12 prime factors of N is even. - If N ≡ {1, 5} mod 6, or if N ≡ {4, 20} mod 24, then the sum of exponents of prime factors of N of the form 7a² + 4ab + 16b² and the sum of exponents of prime factors of N of the form 9a² + 6ab + 13b² do not add up to one. k = 109 Primes p of the form a² + 109b²: Some primes congruent to {1, 5, 9, 21, 25, 29, 45, 49, 61, 73, 81, 89, 93, 97, 105, 109, 113, 121, 125, 129, 137, 145, 157, 169, 173, 189, 193, 197, 209, 213, 217, 221, 225, 233, 245, 249, 253, 261, 281, 289, 293, 301, 305, 349, 353, 361, 365, 373, 393, 401, 405, 409, 421, 429, 433} mod 436, not of the form 5a² + 2ab + 22b² if N is a non-negative integer that can be written as a²+109b², then p × N can be written as a²+109b² if N is a non-negative integer that cannot be written as a²+109b², then p × N cannot be written as a²+109b² Primes of the form 5a² + 2ab + 22b²: Other remaining primes congruent to {1, 5, 9, 21, 25, 29, 45, 49, 61, 73, 81, 89, 93, 97, 105, 113, 121, 125, 129, 137, 145, 157, 169, 173, 189, 193, 197, 209, 213, 217, 221, 225, 233, 245, 249, 253, 261, 281, 289, 293, 301, 305, 349, 353, 361, 365, 373, 393, 401, 405, 409, 421, 429, 433} mod 436, not of the form a²+109b² Primes of the form 10a² + 2ab + 11b²: Some primes congruent to {11, 19, 23, 39, 47, 51, 55, 59, 67, 79, 91, 95, 99, 103, 107, 111, 115, 119, 123, 127, 139, 151, 159, 163, 167, 171, 179, 195, 199, 207, 231, 235, 251, 255, 259, 271, 275, 283, 287, 295, 303, 319, 335, 351, 359, 367, 371, 379, 383, 395, 399, 403, 419, 423} mod 436, not of the form 2a² + 2ab + 55b² Primes of the form 2a² + 2ab + 55b²: Other remaining primes congruent to {2, 11, 19, 23, 39, 47, 51, 55, 59, 67, 79, 91, 95, 99, 103, 107, 111, 115, 119, 123, 127, 139, 151, 159, 163, 167, 171, 179, 195, 199, 207, 231, 235, 251, 255, 259, 271, 275, 283, 287, 295, 303, 319, 335, 351, 359, 367, 371, 379, 383, 395, 399, 403, 419, 423} mod 436, not of the form 10a² + 2ab + 11b² N can be written as a² + 109b² if and only if - N has no prime factors congruent to {3, 7, 13, 15, 17, 27, 31, 33, 35, 37, 41, 43, 53, 57, 63, 65, 69, 71, 75, 77, 83, 85, 87, 101, 117, 131, 133, 135, 141, 143, 147, 149, 153, 155, 161, 165, 175, 177, 181, 183, 185, 187, 191, 201, 203, 205, 211, 215, 219, 223, 227, 229, 237, 239, 241, 243, 247, 257, 263, 265, 267, 269, 273, 277, 279, 285, 291, 297, 299, 307, 309, 311, 313, 315, 317, 321, 323, 325, 329, 331, 333, 337, 339, 341, 343, 345, 347, 355, 357, 363, 369, 375, 377, 381, 385, 387, 389, 391, 397, 407, 411, 413, 415, 417, 425, 427, 431, 435} mod 436 to an odd power. - Sum of exponents of prime factors of N of the form 10a² + 2ab + 11b² and the sum of exponents of prime factors of N of the form 2a² + 2ab + 55b² are of the same parity. - Sum of exponents of prime factors of N of the form 10a² + 2ab + 11b² and the sum of exponents of prime factors of N of the form 5a² + 2ab + 22b² do not add up to one. k = 118 Primes p of the form a² + 118b²: Some primes congruent to {1, 7, 9, 15, 17, 25, 41, 49, 57, 63, 71, 79, 81, 87, 95, 105, 119, 121, 127, 135, 137, 143, 145, 153, 159, 167, 169, 175, 193, 199, 223, 225, 239, 241, 255, 257, 263, 265, 271, 281, 287, 289, 311, 321, 343, 359, 361, 369, 375, 383, 399, 407, 417, 425, 433, 439, 441, 449} mod 472, not of the form 7a² + 2ab + 17b² if N is a non-negative integer that can be written as a²+118b², then p × N can be written as a²+118b² if N is a non-negative integer that cannot be written as a²+118b², then p × N cannot be written as a²+118b² Primes of the form 7a² + 2ab + 17b²: Other remaining primes congruent to {1, 7, 9, 15, 17, 25, 41, 49, 57, 63, 71, 79, 81, 87, 95, 105, 119, 121, 127, 135, 137, 143, 145, 153, 159, 167, 169, 175, 193, 199, 223, 225, 239, 241, 255, 257, 263, 265, 271, 281, 287, 289, 311, 321, 343, 359, 361, 369, 375, 383, 399, 407, 417, 425, 433, 439, 441, 449} mod 472, not of the form a²+118b² Primes of the form 11a² + 10ab + 13b²: Some primes congruent to {11, 13, 37, 43, 61, 67, 69, 77, 83, 91, 93, 99, 101, 109, 115, 117, 131, 141, 149, 155, 157, 165, 173, 179, 187, 195, 211, 219, 221, 227, 229, 235, 259, 267, 269, 275, 283, 291, 301, 309, 325, 333, 339, 347, 349, 365, 387, 397, 419, 421, 427, 437, 443, 445, 451, 453, 467, 469} mod 472, not of the form 2a² + 59b² Primes of the form 2a² + 59b²: Other remaining primes congruent to {2, 11, 13, 37, 43, 59, 61, 67, 69, 77, 83, 91, 93, 99, 101, 109, 115, 117, 131, 141, 149, 155, 157, 165, 173, 179, 187, 195, 211, 219, 221, 227, 229, 235, 259, 267, 269, 275, 283, 291, 301, 309, 325, 333, 339, 347, 349, 365, 387, 397, 419, 421, 427, 437, 443, 445, 451, 453, 467, 469} mod 472, not of the form 11a² + 10ab + 13b² N can be written as a² + 118b² if and only if - N has no prime factors congruent to {3, 5, 19, 21, 23, 27, 29, 31, 33, 35, 39, 45, 47, 51, 53, 55, 65, 73, 75, 85, 89, 97, 103, 107, 111, 113, 123, 125, 129, 133, 139, 147, 151, 161, 163, 171, 181, 183, 185, 189, 191, 197, 201, 203, 205, 207, 209, 213, 215, 217, 231, 233, 237, 243, 245, 247, 249, 251, 253, 261, 273, 277, 279, 285, 293, 297, 299, 303, 305, 307, 313, 315, 317, 319, 323, 327, 329, 331, 335, 337, 341, 345, 351, 353, 355, 357, 363, 367, 371, 373, 377, 379, 381, 385, 389, 391, 393, 395, 401, 403, 405, 409, 411, 415, 423, 429, 431, 435, 447, 455, 457, 459, 461, 463, 465, 471} mod 472 to an odd power. - Sum of exponents of prime factors of N of the form 11a² + 10ab + 13b² and the sum of exponents of prime factors of N of the form 2a² + 59b² are of the same parity. - If N is odd, then the sum of exponents of prime factors of N of the form 11a² + 10ab + 13b² and the sum of exponents of prime factors of N of the form 7a² + 2ab + 17b² do not add up to one.

2013-03-22, 21:43	#115
Raman Noodles "Mr. Tuch" Dec 2007 Chennai, India 10011101001₂ Posts	k = 121 Primes p of the form a² + 121b²: Some primes congruent to {1, 5, 9, 25, 37} mod 44, not of the form 5a² + 4ab + 25b² if N is a non-negative integer that can be written as a²+121b², then p × N can be written as a²+121b² if N is a non-negative integer that cannot be written as a²+121b², then p × N cannot be written as a²+121b² Primes of the form 5a² + 4ab + 25b²: Other remaining primes congruent to {1, 5, 9, 25, 37} mod 44, not of the form a²+121b² Primes of the form 10a² + 6ab + 13b²: Some primes congruent to {13, 17, 21, 29, 41} mod 44, not of the form 2a² + 2ab + 61b² Primes of the form 2a² + 2ab + 61b²: Other remaining primes congruent to {2, 13, 17, 21, 29, 41} mod 44, not of the form 10a² + 6ab + 13b² N can be written as a² + 121b² if and only if - N has no prime factors congruent to {3, 7, 11, 15, 19, 23, 27, 31, 35, 39, 43} mod 44 to an odd power. - If N is not a multiple of 11, then the sum of exponents of prime factors of N of the form 10a² + 6ab + 13b² and the sum of exponents of prime factors of N of the form 2a² + 2ab + 61b² are of the same parity. - If N is not a multiple of 11, then the sum of exponents of prime factors of N of the form 10a² + 6ab + 13b² and the sum of exponents of prime factors of N of the form 5a² + 4ab + 25b² do not add up to one. k = 135 Primes p of the form a² + 135b²: Some primes congruent to {1, 19} mod 30, not of the form 9a² + 12ab + 19b² if N is a non-negative integer that can be written as a²+135b², then p × N can be written as a²+135b² if N is a non-negative integer that cannot be written as a²+135b², then p × N cannot be written as a²+135b² Primes of the form 9a² + 12ab + 19b²: Other remaining primes congruent to {1, 19} mod 30, not of the form a²+135b² Primes of the form 8a² + 2ab + 17b²: Some primes congruent to {17, 23} mod 30, not of the form 5a² + 27b² Primes of the form 5a² + 27b²: Other remaining primes congruent to {5, 17, 23} mod 30, not of the form 8a² + 2ab + 17b² N can be written as a² + 135b² if and only if - N is not congruent to 2 (mod 4). - N is not congruent to {3, 6} (mod 9). - N has no prime factors congruent to {7, 11, 13, 29} mod 30 to an odd power. - Sum of exponents of {2, 3, 5, 17, 23} mod 30 prime factors of N is even. - If N ≡ {1, 5} mod 6, or if N ≡ {4, 20} mod 24, then the sum of exponents of prime factors of N of the form 8a² + 2ab + 17b² and the sum of exponents of prime factors of N of the form 9a² + 12ab + 19b² do not add up to one. - If N ≡ {8, 40} mod 48, then there is atleast one prime factor of N of the form 8a² + 2ab + 17b² or 9a² + 12ab + 19b². k = 142 Primes p of the form a² + 142b²: Some primes congruent to {1, 9, 15, 25, 49, 57, 73, 79, 81, 87, 89, 95, 103, 111, 119, 121, 129, 135, 143, 145, 151, 161, 167, 169, 185, 191, 199, 215, 217, 223, 225, 231, 233, 249, 263, 271, 273, 287, 289, 303, 311, 313, 321, 327, 329, 359, 361, 367, 375, 385, 391, 393, 409, 415, 431, 441, 455, 463, 471, 503, 505, 513, 521, 527, 529, 535, 537, 545, 551, 561} mod 568, not of the form 2a² + 71b² if N is a non-negative integer that can be written as a²+142b², then p × N can be written as a²+142b² if N is a non-negative integer that cannot be written as a²+142b², then p × N cannot be written as a²+142b² Primes of the form 2a² + 71b²: Other remaining primes congruent to {1, 2, 9, 15, 25, 49, 57, 71, 73, 79, 81, 87, 89, 95, 103, 111, 119, 121, 129, 135, 143, 145, 151, 161, 167, 169, 185, 191, 199, 215, 217, 223, 225, 231, 233, 249, 263, 271, 273, 287, 289, 303, 311, 313, 321, 327, 329, 359, 361, 367, 375, 385, 391, 393, 409, 415, 431, 441, 455, 463, 471, 503, 505, 513, 521, 527, 529, 535, 537, 545, 551, 561} mod 568, not of the form a²+142b² Primes of the form 11a² + 2ab + 13b²: All primes congruent to {11, 13, 21, 35, 51, 53, 59, 61, 67, 69, 85, 93, 99, 115, 117, 123, 133, 139, 141, 149, 155, 163, 165, 173, 181, 189, 195, 197, 203, 205, 211, 227, 235, 259, 269, 275, 283, 291, 301, 307, 315, 317, 323, 325, 331, 339, 347, 349, 381, 389, 397, 411, 421, 437, 443, 459, 461, 467, 477, 485, 491, 493, 523, 525, 531, 539, 541, 549, 563, 565} mod 568 N can be written as a² + 142b² if and only if - N has no prime factors congruent to {3, 5, 7, 17, 19, 23, 27, 29, 31, 33, 37, 39, 41, 43, 45, 47, 55, 63, 65, 75, 77, 83, 91, 97, 101, 105, 107, 109, 113, 125, 127, 131, 137, 147, 153, 157, 159, 171, 175, 177, 179, 183, 187, 193, 201, 207, 209, 219, 221, 229, 237, 239, 241, 243, 245, 247, 251, 253, 255, 257, 261, 265, 267, 277, 279, 281, 285, 293, 295, 297, 299, 305, 309, 319, 333, 335, 337, 341, 343, 345, 351, 353, 357, 363, 365, 369, 371, 373, 377, 379, 383, 387, 395, 399, 401, 403, 405, 407, 413, 417, 419, 423, 425, 427, 429, 433, 435, 439, 445, 447, 449, 451, 453, 457, 465, 469, 473, 475, 479, 481, 483, 487, 489, 495, 499, 501, 507, 509, 511, 515, 517, 519, 533, 543, 547, 553, 555, 557, 559, 567} mod 568 to an odd power. - Sum of exponents of {11, 13, 21, 35, 51, 53, 59, 61, 67, 69, 85, 93, 99, 115, 117, 123, 133, 139, 141, 149, 155, 163, 165, 173, 181, 189, 195, 197, 203, 205, 211, 227, 235, 259, 269, 275, 283, 291, 301, 307, 315, 317, 323, 325, 331, 339, 347, 349, 381, 389, 397, 411, 421, 437, 443, 459, 461, 467, 477, 485, 491, 493, 523, 525, 531, 539, 541, 549, 563, 565} mod 568 prime factors of N is even. - If N has no prime factors congruent to {11, 13, 21, 35, 51, 53, 59, 61, 67, 69, 85, 93, 99, 115, 117, 123, 133, 139, 141, 149, 155, 163, 165, 173, 181, 189, 195, 197, 203, 205, 211, 227, 235, 259, 269, 275, 283, 291, 301, 307, 315, 317, 323, 325, 331, 339, 347, 349, 381, 389, 397, 411, 421, 437, 443, 459, 461, 467, 477, 485, 491, 493, 523, 525, 531, 539, 541, 549, 563, 565} mod 568, then sum of exponents of prime factors of N of the form 2a² + 71b² is even. k = 147 Primes p of the form a² + 147b²: Some primes congruent to {1, 16, 25} mod 21, not of the form 4a² + 2ab + 37b² if N is a non-negative integer that can be written as a²+147b², then p × N can be written as a²+147b² if N is a non-negative integer that cannot be written as a²+147b², then p × N cannot be written as a²+147b² Primes of the form 4a² + 2ab + 37b²: Other remaining primes congruent to {1, 16, 25} mod 21, not of the form a²+147b² Primes of the form 12a² + 6ab + 13b²: Some primes congruent to {10, 13, 19} mod 21, not of the form 3a² + 49b² Primes of the form 3a² + 49b²: Other remaining primes congruent to {3, 10, 13, 19} mod 21, not of the form 12a² + 6ab + 13b² N can be written as a² + 147b² if and only if - N is not congruent to {7, 14, 21, 28, 35, 42} (mod 49). - N has no prime factors congruent to {2, 5, 8, 11, 17, 20} mod 21 to an odd power. - If N is not a multiple of 7, then the sum of exponents of prime factors of N of the form 12a² + 6ab + 13b² and the sum of exponents of prime factors of N of the form 3a² + 49b² are of the same parity. - If N is co-prime to 14, then the sum of exponents of prime factors of N of the form 12a² + 6ab + 13b² and the sum of exponents of prime factors of N of the form 4a² + 2ab + 37b² do not add up to one. k = 148 Primes p of the form a² + 148b²: Some primes congruent to {1, 9, 21, 25, 33, 41, 49, 53, 65, 73, 77, 81, 85, 101, 121, 137, 141, 145} mod 148, not of the form 4a² + 37b² if N is a non-negative integer that can be written as a²+148b², then p × N can be written as a²+148b² if N is a non-negative integer that cannot be written as a²+148b², then p × N cannot be written as a²+148b² Primes of the form 4a² + 37b²: Other remaining primes congruent to {1, 9, 21, 25, 33, 37, 41, 49, 53, 65, 73, 77, 81, 85, 101, 121, 137, 141, 145} mod 148, not of the form a²+148b² Primes of the form 8a² + 4ab + 19b²: All primes congruent to {15, 19, 23, 31, 35, 39, 43, 51, 55, 59, 79, 87, 91, 103, 119, 131, 135, 143} mod 148 N can be written as a² + 148b² if and only if - N is not congruent to 2 (mod 4). - N has no prime factors congruent to {3, 5, 7, 11, 13, 17, 27, 29, 45, 47, 57, 61, 63, 67, 69, 71, 75, 83, 89, 93, 95, 97, 99, 105, 107, 109, 113, 115, 117, 123, 125, 127, 129, 133, 139, 147} mod 148 to an odd power. - Sum of exponents of {2, 15, 19, 23, 31, 35, 39, 43, 51, 55, 59, 79, 87, 91, 103, 119, 131, 135, 143} mod 148 prime factors of N is even. - If N has no prime factors congruent to {2, 15, 19, 23, 31, 35, 39, 43, 51, 55, 59, 79, 87, 91, 103, 119, 131, 135, 143} mod 148, then sum of exponents of prime factors of N of the form 4a² + 37b² is even.

2013-03-23, 23:34	#119
Raman Noodles "Mr. Tuch" Dec 2007 Chennai, India 2351₈ Posts	k = 151 Primes p of the form a² + 151b²: Some primes congruent to {1, 5, 9, 11, 17, 19, 21, 25, 29, 31, 37, 39, 43, 45, 47, 49, 55, 59, 69, 81, 85, 91, 95, 97, 99, 103, 105, 121, 123, 125, 127, 137, 139, 145, 151, 153, 155, 159, 161, 167, 169, 171, 173, 183, 185, 187, 189, 191, 193, 195, 201, 209, 213, 215, 219, 223, 225, 227, 229, 231, 235, 237, 239, 241, 245, 249, 251, 261, 267, 269, 275, 279, 287, 289, 295, 299} mod 302, neither of the form 5a² + 4ab + 31b² nor 11a² + 10ab + 16b² nor 8a² + 2ab + 19b² if N is a non-negative integer that can be written as a²+151b², then p × N can be written as a²+151b² if N is a non-negative integer that cannot be written as a²+151b², then p × N cannot be written as a²+151b² Primes of the form 5a² + 4ab + 31b²: Some more primes congruent to {1, 5, 9, 11, 17, 19, 21, 25, 29, 31, 37, 39, 43, 45, 47, 49, 55, 59, 69, 81, 85, 91, 95, 97, 99, 103, 105, 121, 123, 125, 127, 137, 139, 145, 153, 155, 159, 161, 167, 169, 171, 173, 183, 185, 187, 189, 191, 193, 195, 201, 209, 213, 215, 219, 223, 225, 227, 229, 231, 235, 237, 239, 241, 245, 249, 251, 261, 267, 269, 275, 279, 287, 289, 295, 299} mod 302, neither of the form a²+151b² nor 11a² + 10ab + 16b² nor 8a² + 2ab + 19b² Primes of the form 11a² + 10ab + 16b²: Some more primes congruent to {1, 5, 9, 11, 17, 19, 21, 25, 29, 31, 37, 39, 43, 45, 47, 49, 55, 59, 69, 81, 85, 91, 95, 97, 99, 103, 105, 121, 123, 125, 127, 137, 139, 145, 153, 155, 159, 161, 167, 169, 171, 173, 183, 185, 187, 189, 191, 193, 195, 201, 209, 213, 215, 219, 223, 225, 227, 229, 231, 235, 237, 239, 241, 245, 249, 251, 261, 267, 269, 275, 279, 287, 289, 295, 299} mod 302, neither of the form a²+151b² nor 5a² + 4ab + 31b² nor 8a² + 2ab + 19b² Primes of the form 8a² + 2ab + 19b²: Other remaining primes congruent to {1, 5, 9, 11, 17, 19, 21, 25, 29, 31, 37, 39, 43, 45, 47, 49, 55, 59, 69, 81, 85, 91, 95, 97, 99, 103, 105, 121, 123, 125, 127, 137, 139, 145, 153, 155, 159, 161, 167, 169, 171, 173, 183, 185, 187, 189, 191, 193, 195, 201, 209, 213, 215, 219, 223, 225, 227, 229, 231, 235, 237, 239, 241, 245, 249, 251, 261, 267, 269, 275, 279, 287, 289, 295, 299} mod 302, neither of the form a²+151b² nor 5a² + 4ab + 31b² nor 11a² + 10ab + 16b² N can be written as a² + 151b² if and only if - N is not congruent to 2 (mod 4). - N has no prime factors congruent to {3, 7, 13, 15, 23, 27, 33, 35, 41, 51, 53, 57, 61, 63, 65, 67, 71, 73, 75, 77, 79, 83, 87, 89, 93, 101, 107, 109, 111, 113, 115, 117, 119, 129, 131, 133, 135, 141, 143, 147, 149, 157, 163, 165, 175, 177, 179, 181, 197, 199, 203, 205, 207, 211, 217, 221, 233, 243, 247, 253, 255, 257, 259, 263, 265, 271, 273, 277, 281, 283, 285, 291, 293, 297, 301} mod 302 to an odd power. - If N is odd or N ≡ 4 (mod 8), then the (sum of exponents of prime factors of N of the form 5a² + 4ab + 31b², sum of exponents of prime factors of N of the form 11a² + 10ab + 16b², sum of exponents of prime factors of N of the form 8a² + 2ab + 19b²) is not (1,0,0) or (0,1,0) or (0,0,1) or (1,1,0) or (1,0,1) or (0,1,1) or (3,0,0) or (0,3,0) or (0,0,3) or (2,1,0) or (0,2,1) or (1,0,2) or (1,3,0) or (0,1,3) or (3,0,1) or (5,0,0) or (0,5,0). - If N ≡ 8 (mod 16), then the (sum of exponents of prime factors of N of the form 5a² + 4ab + 31b², sum of exponents of prime factors of N of the form 11a² + 10ab + 16b², sum of exponents of prime factors of N of the form 8a² + 2ab + 19b²) is not (0,0,0) or (1,0,0) or (0,1,0) or (1,0,1) or (0,2,0) or (0,0,2) or (0,1,2) or (3,0,0). - If N ≡ 16 (mod 32), then the (sum of exponents of prime factors of N of the form 5a² + 4ab + 31b², sum of exponents of prime factors of N of the form 11a² + 10ab + 16b², sum of exponents of prime factors of N of the form 8a² + 2ab + 19b²) is not (1,0,0) or (0,0,1) or (0,1,1) or (0,0,3). - If N ≡ 32 (mod 64), then the (sum of exponents of prime factors of N of the form 5a² + 4ab + 31b², sum of exponents of prime factors of N of the form 11a² + 10ab + 16b², sum of exponents of prime factors of N of the form 8a² + 2ab + 19b²) is not (0,0,0) or (0,1,0) or (0,0,2). - If N ≡ 64 (mod 128), then the (sum of exponents of prime factors of N of the form 5a² + 4ab + 31b², sum of exponents of prime factors of N of the form 11a² + 10ab + 16b², sum of exponents of prime factors of N of the form 8a² + 2ab + 19b²) is not (0,0,1). - If N ≡ 128 (mod 256), then the (sum of exponents of prime factors of N of the form 5a² + 4ab + 31b², sum of exponents of prime factors of N of the form 11a² + 10ab + 16b², sum of exponents of prime factors of N of the form 8a² + 2ab + 19b²) is not (0,0,0). k = 157 Primes p of the form a² + 157b²: Some primes congruent to {1, 9, 13, 17, 25, 33, 37, 49, 57, 81, 89, 93, 101, 105, 109, 113, 117, 121, 141, 145, 153, 157, 161, 169, 173, 193, 197, 201, 205, 209, 213, 221, 225, 233, 257, 265, 277, 281, 289, 297, 301, 305, 313, 317, 325, 333, 341, 345, 349, 353, 361, 365, 381, 385, 389, 413, 425, 429, 441, 457, 461, 481, 485, 501, 513, 517, 529, 553, 557, 561, 577, 581, 589, 593, 597, 601, 609, 617, 625} mod 628, not of the form 13a² + 10ab + 14b² if N is a non-negative integer that can be written as a²+157b², then p × N can be written as a²+157b² if N is a non-negative integer that cannot be written as a²+157b², then p × N cannot be written as a²+157b² Primes of the form 13a² + 10ab + 14b²: Other remaining primes congruent to {1, 9, 13, 17, 25, 33, 37, 49, 57, 81, 89, 93, 101, 105, 109, 113, 117, 121, 141, 145, 153, 161, 169, 173, 193, 197, 201, 205, 209, 213, 221, 225, 233, 257, 265, 277, 281, 289, 297, 301, 305, 313, 317, 325, 333, 341, 345, 349, 353, 361, 365, 381, 385, 389, 413, 425, 429, 441, 457, 461, 481, 485, 501, 513, 517, 529, 553, 557, 561, 577, 581, 589, 593, 597, 601, 609, 617, 625} mod 628, not of the form a²+157b² Primes of the form 7a² + 4ab + 23b²: Some primes congruent to {7, 15, 23, 43, 55, 59, 63, 79, 83, 87, 91, 95, 103, 107, 119, 123, 131, 135, 139, 151, 155, 159, 163, 175, 179, 183, 191, 195, 207, 211, 219, 223, 227, 231, 235, 251, 255, 259, 271, 291, 299, 307, 319, 335, 343, 355, 359, 367, 375, 379, 383, 387, 391, 399, 411, 439, 443, 447, 451, 463, 479, 491, 495, 499, 503, 531, 543, 551, 555, 559, 563, 567, 575, 583, 587, 599, 607, 623} mod 628, not of the form 2a² + 2ab + 79b² Primes of the form 2a² + 2ab + 79b²: Other remaining primes congruent to {2, 7, 15, 23, 43, 55, 59, 63, 79, 83, 87, 91, 95, 103, 107, 119, 123, 131, 135, 139, 151, 155, 159, 163, 175, 179, 183, 191, 195, 207, 211, 219, 223, 227, 231, 235, 251, 255, 259, 271, 291, 299, 307, 319, 335, 343, 355, 359, 367, 375, 379, 383, 387, 391, 399, 411, 439, 443, 447, 451, 463, 479, 491, 495, 499, 503, 531, 543, 551, 555, 559, 563, 567, 575, 583, 587, 599, 607, 623} mod 628, not of the form 7a² + 4ab + 23b² N can be written as a² + 157b² if and only if - N has no prime factors congruent to {3, 5, 11, 19, 21, 27, 29, 31, 35, 39, 41, 45, 47, 51, 53, 61, 65, 67, 69, 71, 73, 75, 77, 85, 97, 99, 111, 115, 125, 127, 129, 133, 137, 143, 147, 149, 165, 167, 171, 177, 181, 185, 187, 189, 199, 203, 215, 217, 229, 237, 239, 241, 243, 245, 247, 249, 253, 261, 263, 267, 269, 273, 275, 279, 283, 285, 287, 293, 295, 303, 309, 311, 315, 321, 323, 327, 329, 331, 337, 339, 347, 351, 357, 363, 369, 371, 373, 377, 393, 395, 397, 401, 403, 405, 407, 409, 415, 417, 419, 421, 423, 427, 431, 433, 435, 437, 445, 449, 453, 455, 459, 465, 467, 469, 473, 475, 477, 483, 487, 489, 493, 497, 505, 507, 509, 511, 515, 519, 521, 523, 525, 527, 533, 535, 537, 539, 541, 545, 547, 549, 565, 569, 571, 573, 579, 585, 591, 595, 603, 605, 611, 613, 615, 619, 621, 627} mod 628 to an odd power. - Sum of exponents of prime factors of N of the form 7a² + 4ab + 23b² and the sum of exponents of prime factors of N of the form 2a² + 2ab + 79b² are of the same parity. - Sum of exponents of prime factors of N of the form 7a² + 4ab + 23b² and the sum of exponents of prime factors of N of the form 13a² + 10ab + 14b² do not add up to one.

2013-03-23, 23:37	#121
Raman Noodles "Mr. Tuch" Dec 2007 Chennai, India 10011101001₂ Posts	k = 175 Primes p of the form a² + 175b²: Some primes congruent to {1, 9, 11, 29, 39, 51} mod 70, not of the form 11a² + 2ab + 16b² if N is a non-negative integer that can be written as a²+175b², then p × N can be written as a²+175b² if N is a non-negative integer that cannot be written as a²+175b², then p × N cannot be written as a²+175b² Primes of the form 11a² + 2ab + 16b²: Other remaining primes congruent to {1, 9, 11, 29, 39, 51} mod 70, not of the form a²+175b² Primes of the form 8a² + 6ab + 23b²: Some primes congruent to {23, 37, 43, 53, 57, 67} mod 70, not of the form 7a² + 25b² Primes of the form 7a² + 25b²: Other remaining primes congruent to {7, 23, 37, 43, 53, 57, 67} mod 70, not of the form 8a² + 6ab + 23b² N can be written as a² + 175b² if and only if - N is not congruent to 2 (mod 4). - N has no prime factors congruent to {3, 5, 13, 17, 19, 27, 31, 33, 41, 47, 59, 61, 69} mod 70 to an odd power. - If N is not a multiple of 5, and the highest power of 2 dividing N is an even number (or zero), then the sum of exponents of prime factors of N of the form 8a² + 6ab + 23b² and the sum of exponents of prime factors of N of the form 7a² + 25b² are of the same parity. - If N is not a multiple of 5, and the highest power of 2 dividing N is an odd number that is three or more, then the sum of exponents of prime factors of N of the form 8a² + 6ab + 23b² and the sum of exponents of prime factors of N of the form 7a² + 25b² are of different parity. - If N ≡ {1, 3, 7, 9} (mod 10) or if N ≡ {4, 12, 28, 36} (mod 40), then the sum of exponents of prime factors of N of the form 8a² + 6ab + 23b² and the sum of exponents of prime factors of N of the form 11a² + 2ab + 16b² do not add up to one. - If N ≡ {8, 24, 56, 72} (mod 80), then there is atleast one prime factor of N of the form 8a² + 6ab + 23b² or 11a² + 2ab + 16b². k = 187 Primes p of the form a² + 187b²: Some primes congruent to {1, 4, 9, 15, 16, 25, 26, 36, 38, 42, 47, 49, 53, 59, 60, 64, 67, 69, 70, 81, 86, 89, 93, 100, 103, 104, 111, 115, 135, 137, 144, 152, 155, 157, 166, 168, 169, 174, 179, 185} mod 187, not of the form 4a² + 2ab + 47b² if N is a non-negative integer that can be written as a²+187b², then p × N can be written as a²+187b² if N is a non-negative integer that cannot be written as a²+187b², then p × N cannot be written as a²+187b² Primes of the form 4a² + 2ab + 47b²: Other remaining primes congruent to {1, 4, 9, 15, 16, 25, 26, 36, 38, 42, 47, 49, 53, 59, 60, 64, 67, 69, 70, 81, 86, 89, 93, 100, 103, 104, 111, 115, 135, 137, 144, 152, 155, 157, 166, 168, 169, 174, 179, 185} mod 187, not of the form a²+187b² Primes of the form 7a² + 6ab + 28b²: Some primes congruent to {6, 7, 10, 24, 28, 29, 39, 40, 41, 46, 54, 57, 61, 62, 63, 65, 73, 74, 79, 90, 95, 96, 105, 107, 109, 112, 116, 129, 131, 139, 142, 150, 156, 160, 164, 167, 173, 175, 182, 184} mod 187, not of the form 11a² + 17b² Primes of the form 11a² + 17b²: Other remaining primes congruent to {6, 7, 10, 11, 17, 24, 28, 29, 39, 40, 41, 46, 54, 57, 61, 62, 63, 65, 73, 74, 79, 90, 95, 96, 105, 107, 109, 112, 116, 129, 131, 139, 142, 150, 156, 160, 164, 167, 173, 175, 182, 184} mod 187, not of the form 7a² + 6ab + 28b² N can be written as a² + 187b² if and only if - N has no prime factors congruent to {2, 3, 5, 8, 12, 13, 14, 18, 19, 20, 21, 23, 27, 30, 31, 32, 35, 37, 43, 45, 48, 50, 52, 56, 58, 71, 72, 75, 76, 78, 80, 82, 83, 84, 87, 91, 92, 94, 97, 98, 101, 106, 108, 113, 114, 117, 118, 120, 122, 123, 124, 125, 126, 127, 128, 130, 133, 134, 138, 140, 141, 145, 146, 147, 148, 149, 151, 158, 159, 161, 162, 163, 171, 172, 177, 178, 180, 181, 183, 186} mod 187 to an odd power. - Sum of exponents of prime factors of N of the form 7a² + 6ab + 28b² and the sum of exponents of prime factors of N of the form 11a² + 17b² are of the same parity. - If N is odd, then the sum of exponents of prime factors of N of the form 7a² + 6ab + 28b² and the sum of exponents of prime factors of N of the form 4a² + 2ab + 47b² do not add up to one. k = 193 Primes p of the form a² + 193b²: Some primes congruent to {1, 9, 21, 25, 49, 65, 69, 81, 85, 93, 97, 101, 109, 121, 129, 137, 145, 157, 161, 165, 169, 177, 181, 185, 189, 193, 197, 201, 205, 209, 217, 221, 225, 229, 241, 249, 257, 265, 277, 285, 289, 293, 301, 305, 317, 321, 337, 361, 365, 377, 385, 389, 393, 409, 413, 417, 429, 441, 445, 449, 453, 461, 469, 481, 493, 517, 525, 529, 533, 537, 561, 565, 573, 577, 581, 585, 593, 597, 621, 625, 629, 633, 641, 665, 677, 689, 697, 705, 709, 713, 717, 729, 741, 745, 749, 765, 769} mod 772, not of the form 2a² + 2ab + 97b² if N is a non-negative integer that can be written as a²+193b², then p × N can be written as a²+193b² if N is a non-negative integer that cannot be written as a²+193b², then p × N cannot be written as a²+193b² Primes of the form 2a² + 2ab + 97b²: Other remaining primes congruent to {1, 2, 9, 21, 25, 49, 65, 69, 81, 85, 93, 97, 101, 109, 121, 129, 137, 145, 157, 161, 165, 169, 177, 181, 185, 189, 197, 201, 205, 209, 217, 221, 225, 229, 241, 249, 257, 265, 277, 285, 289, 293, 301, 305, 317, 321, 337, 361, 365, 377, 385, 389, 393, 409, 413, 417, 429, 441, 445, 449, 453, 461, 469, 481, 493, 517, 525, 529, 533, 537, 561, 565, 573, 577, 581, 585, 593, 597, 621, 625, 629, 633, 641, 665, 677, 689, 697, 705, 709, 713, 717, 729, 741, 745, 749, 765, 769} mod 772, not of the form a²+193b² Primes of the form 11a² + 8ab + 19b²: All primes congruent to {11, 15, 19, 35, 39, 47, 51, 71, 79, 87, 91, 99, 103, 111, 115, 119, 123, 127, 135, 155, 159, 163, 167, 171, 183, 203, 215, 219, 223, 227, 231, 251, 259, 263, 267, 271, 275, 283, 287, 295, 299, 307, 315, 335, 339, 347, 351, 367, 371, 375, 391, 399, 403, 415, 419, 423, 427, 431, 439, 443, 447, 459, 463, 475, 491, 499, 503, 511, 519, 527, 535, 539, 559, 599, 619, 623, 631, 639, 647, 655, 659, 667, 683, 695, 699, 711, 715, 719, 727, 731, 735, 739, 743, 755, 759, 767} mod 772 N can be written as a² + 193b² if and only if - N has no prime factors congruent to {3, 5, 7, 13, 17, 23, 27, 29, 31, 33, 37, 41, 43, 45, 53, 55, 57, 59, 61, 63, 67, 73, 75, 77, 83, 89, 95, 105, 107, 113, 117, 125, 131, 133, 139, 141, 143, 147, 149, 151, 153, 173, 175, 179, 187, 191, 195, 199, 207, 211, 213, 233, 235, 237, 239, 243, 245, 247, 253, 255, 261, 269, 273, 279, 281, 291, 297, 303, 309, 311, 313, 319, 323, 325, 327, 329, 331, 333, 341, 343, 345, 349, 353, 355, 357, 359, 363, 369, 373, 379, 381, 383, 387, 395, 397, 401, 405, 407, 411, 421, 425, 433, 435, 437, 451, 455, 457, 465, 467, 471, 473, 477, 479, 483, 485, 487, 489, 495, 497, 501, 505, 507, 509, 513, 515, 521, 523, 531, 541, 543, 545, 547, 549, 551, 553, 555, 557, 563, 567, 569, 571, 575, 583, 587, 589, 591, 595, 601, 603, 605, 607, 609, 611, 613, 615, 617, 627, 635, 637, 643, 645, 649, 651, 653, 657, 661, 663, 669, 671, 673, 675, 679, 681, 685, 687, 691, 693, 701, 703, 707, 721, 723, 725, 733, 737, 747, 751, 753, 757, 761, 763, 771} mod 772 to an odd power. - Sum of exponents of {11, 15, 19, 35, 39, 47, 51, 71, 79, 87, 91, 99, 103, 111, 115, 119, 123, 127, 135, 155, 159, 163, 167, 171, 183, 203, 215, 219, 223, 227, 231, 251, 259, 263, 267, 271, 275, 283, 287, 295, 299, 307, 315, 335, 339, 347, 351, 367, 371, 375, 391, 399, 403, 415, 419, 423, 427, 431, 439, 443, 447, 459, 463, 475, 491, 499, 503, 511, 519, 527, 535, 539, 559, 599, 619, 623, 631, 639, 647, 655, 659, 667, 683, 695, 699, 711, 715, 719, 727, 731, 735, 739, 743, 755, 759, 767} mod 772 prime factors of N is even. - If N has no prime factors congruent to {11, 15, 19, 35, 39, 47, 51, 71, 79, 87, 91, 99, 103, 111, 115, 119, 123, 127, 135, 155, 159, 163, 167, 171, 183, 203, 215, 219, 223, 227, 231, 251, 259, 263, 267, 271, 275, 283, 287, 295, 299, 307, 315, 335, 339, 347, 351, 367, 371, 375, 391, 399, 403, 415, 419, 423, 427, 431, 439, 443, 447, 459, 463, 475, 491, 499, 503, 511, 519, 527, 535, 539, 559, 599, 619, 623, 631, 639, 647, 655, 659, 667, 683, 695, 699, 711, 715, 719, 727, 731, 735, 739, 743, 755, 759, 767} mod 772, then sum of exponents of prime factors of N of the form 2a² + 2ab + 97b² is even.