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

[Raman]

For a given k,
it is being a good question in order to ask, how many, (what all), quadratic polynomial forms are being needed in order to generate / represent / write with in
all the prime numbers that are being congruent to x² (mod 4k), x²+k (mod 4k).

The following answer is being given below as follows:-> , All of the following polynomials will have got with in a discriminant of -4k,
by itself, over thereby,

The following Program Output, is being given below, as follows,

k = 1: a² + b²
k = 2: a² + 2b²
k = 3: a² + 3b²
k = 4: a² + 4b²
k = 5: a² + 5b²
k = 6: a² + 6b²
k = 7: a² + 7b²
k = 8: a² + 8b²
k = 9: a² + 9b²
k = 10: a² + 10b²
k = 11: a² + 11b², 3a² + 2ab + 4b²
k = 12: a² + 12b²
k = 13: a² + 13b²
k = 14: a² + 14b², 2a² + 7b²
k = 15: a² + 15b²
k = 16: a² + 16b²
k = 17: a² + 17b², 2a² + 2ab + 9b²
k = 18: a² + 18b²
k = 19: a² + 19b², 4a² + 2ab + 5b²
k = 20: a² + 20b², 4a² + 5b²
k = 21: a² + 21b²
k = 22: a² + 22b²
k = 23: a² + 23b², 3a² + 2ab + 8b²
k = 24: a² + 24b²
k = 25: a² + 25b²
k = 26: a² + 26b², 3a² + 2ab + 9b²
k = 27: a² + 27b², 4a² + 2ab + 7b²
k = 28: a² + 28b²
k = 29: a² + 29b², 5a² + 2ab + 6b²
k = 30: a² + 30b²
k = 31: a² + 31b², 5a² + 4ab + 7b²
k = 32: a² + 32b², 4a² + 4ab + 9b²
k = 33: a² + 33b²
k = 34: a² + 34b², 2a² + 17b²
k = 35: a² + 35b², 4a² + 6ab + 11b²
k = 36: a² + 36b², 4a² + 9b²
k = 37: a² + 37b²
k = 38: a² + 38b², 6a² + 4ab + 7b²
k = 39: a² + 39b², 3a² + 13b²
k = 40: a² + 40b²
k = 41: a² + 41b², 2a² + 2ab + 21b², 5a² + 4ab + 9b²
k = 42: a² + 42b²
k = 43: a² + 43b², 4a² + 2ab + 11b²
k = 44: a² + 44b², 5a² + 2ab + 9b²
k = 45: a² + 45b²
k = 46: a² + 46b², 2a² + 23b²
k = 47: a² + 47b², 3a² + 2ab + 16b², 7a² + 6ab + 8b²
k = 48: a² + 48b²
k = 49: a² + 49b², 2a² + 2ab + 25b²
k = 50: a² + 50b², 6a² + 8ab + 11b²
k = 51: a² + 51b², 4a² + 2ab + 13b²
k = 52: a² + 52b², 4a² + 13b²
k = 53: a² + 53b², 6a² + 10ab + 13b²
k = 54: a² + 54b², 7a² + 6ab + 9b²
k = 55: a² + 55b², 5a² + 11b²
k = 56: a² + 56b², 8a² + 8ab + 9b²
k = 57: a² + 57b²
k = 58: a² + 58b²
k = 59: a² + 59b², 3a² + 2ab + 20b², 5a² + 2ab + 12b², 7a² + 4ab + 9b², 4a² + 6ab + 17b²
k = 60: a² + 60b²
k = 61: a² + 61b², 5a² + 4ab + 13b²
k = 62: a² + 62b², 2a² + 31b², 7a² + 2ab + 9b²
k = 63: a² + 63b², 7a² + 9b²
k = 64: a² + 64b², 4a² + 4ab + 17b²
k = 65: a² + 65b², 9a² + 10ab + 10b²
k = 66: a² + 66b², 3a² + 22b²
k = 67: a² + 67b², 4a² + 2ab + 17b²
k = 68: a² + 68b², 8a² + 12ab + 13b², 4a² + 17b²
k = 69: a² + 69b², 6a² + 6ab + 13b²
k = 70: a² + 70b²
k = 71: a² + 71b², 3a² + 2ab + 24b², 5a² + 4ab + 15b², 8a² + 2ab + 9b²
k = 72: a² + 72b²
k = 73: a² + 73b², 2a² + 2ab + 37b²
k = 74: a² + 74b², 3a² + 2ab + 25b², 9a² + 10ab + 11b²
k = 75: a² + 75b², 4a² + 2ab + 19b²
k = 76: a² + 76b², 5a² + 4ab + 16b²
k = 77: a² + 77b², 9a² + 14ab + 14b²
k = 78: a² + 78b²
k = 79: a² + 79b², 5a² + 2ab + 16b², 8a² + 6ab + 11b²
k = 80: a² + 80b², 9a² + 16ab + 16b²
k = 81: a² + 81b², 9a² + 12ab + 13b²
k = 82: a² + 82b², 2a² + 41b²
k = 83: a² + 83b², 3a² + 2ab + 28b², 7a² + 2ab + 12b², 9a² + 8ab + 11b², 4a² + 6ab + 23b²
k = 84: a² + 84b², 4a² + 21b²
k = 85: a² + 85b²
k = 86: a² + 86b², 6a² + 8ab + 17b², 9a² + 4ab + 10b²
k = 87: a² + 87b², 7a² + 4ab + 13b²
k = 88: a² + 88b²
k = 89: a² + 89b², 2a² + 2ab + 45b², 5a² + 2ab + 18b², 9a² + 16ab + 17b²
k = 90: a² + 90b², 9a² + 10b²
k = 91: a² + 91b², 4a² + 2ab + 23b²
k = 92: a² + 92b², 9a² + 10ab + 13b²
k = 93: a² + 93b²
k = 94: a² + 94b², 2a² + 47b², 7a² + 4ab + 14b²
k = 95: a² + 95b², 5a² + 19b², 9a² + 4ab + 11b²
k = 96: a² + 96b², 4a² + 4ab + 25b²
k = 97: a² + 97b², 2a² + 2ab + 49b²
k = 98: a² + 98b², 2a² + 49b², 9a² + 2ab + 11b²
k = 99: a² + 99b², 4a² + 2ab + 25b²
k = 100: a² + 100b², 4a² + 25b²

[Raman]

enough

Being difficult to produce manually the necessary and the sufficient conditions for the class number 8
a²+kb² forms representations, then I had decided out to write off the program for the automatic generation
of the classes that which can combine together when the sum of exponents of these classes of the prime
factors of N are being odd simultaneously, the sum of exponents of the rest of the classes of the prime
factors of N are being all even individually, separately
by itself, over thereby,
using

enough

for the number N, in general, within
whether prime or composite, in order to be,
else, or otherwise
being written into a²+kb² forms, ,
, ,
, , following, , enough, , as, , follows, , below, , enough, ,
, ,
, ,
for the condition being, needed, in, to, be, being,

enough

k = 105

Primes p of the form a²+105b²: All primes congruent to [1, 109, 121, 169, 289, 361] mod 420.
if N is a non-negative integer that can be written as a²+105b², then p × N can be written as a²+105b²
if N is a non-negative integer that cannot be written as a²+105b², then p × N cannot be written as a²+105b²

Class A primes: All primes congruent to [2, 53, 113, 137, 197, 233, 317] mod 420
Class B primes: All primes congruent to [3, 47, 83, 143, 167, 227, 383] mod 420
Class C primes: All primes congruent to [5, 41, 89, 101, 209, 269, 341] mod 420
Class D primes: All primes congruent to [7, 43, 67, 127, 163, 247, 403] mod 420
Class E primes: All primes congruent to [11, 71, 179, 191, 239, 359] mod 420
Class F primes: All primes congruent to [13, 73, 97, 157, 313, 397] mod 420
Class G primes: All primes congruent to [19, 31, 139, 199, 271, 391] mod 420

N can be written as a²+105b² if and only if
- N has no prime factors congruent to [17, 23, 29, 37, 59, 61, 79, 103, 107, 131, 149, 151, 173, 181, 187, 193, 211, 221, 223, 229, 241, 251, 253, 257, 263, 277, 281, 283, 293, 299, 307, 311, 319, 323, 331, 337, 347, 349, 353, 367, 373, 377, 379, 389, 401, 407, 409, 419] mod 420 to an odd power.
- Sum of exponents of the prime factors of N of the classes A,B,C,D,E,F,G are all simultaneously odd, or the sum of exponents of the prime factors of N of the classes A,B,C,D,E,F,G are all simultaneously even, or the sum of exponents of the prime factors of N of these following classes {B,C,D or A,B,C,E or A,D,E or A,C,F or A,B,D,F or B,E,F or C,D,E,F or A,B,G or A,C,D,G or C,E,G or B,D,E,G or B,C,F,G or D,F,G or A,E,F,G} are all simultaneously odd, with the sum of exponents of the prime factors of N of the rest of the classes are all simultaneously even.

k = 120

Primes p of the form a²+120b²: All primes congruent to [1, 49, 121, 169, 241, 289, 361, 409] mod 480.

if N is a non-negative integer that can be written as a²+120b², then p × N can be written as a²+120b²
if N is a non-negative integer that cannot be written as a²+120b², then p × N cannot be written as a²+120b²

Class A primes: All primes congruent to [3, 43, 67, 163, 187, 283, 307, 403, 427] mod 480
Class B primes: All primes congruent to [5, 29, 101, 149, 221, 269, 341, 389, 461] mod 480
Class C primes: All primes congruent to [11, 59, 131, 179, 251, 299, 371, 419] mod 480
Class D primes: All primes congruent to [13, 37, 133, 157, 253, 277, 373, 397] mod 480
Class E primes: All primes congruent to [17, 113, 137, 233, 257, 353, 377, 473] mod 480
Class F primes: All primes congruent to [23, 47, 143, 167, 263, 287, 383, 407] mod 480
Class G primes: All primes congruent to [31, 79, 151, 199, 271, 319, 391, 439] mod 480

N can be written as a²+120b² if and only if
- N is not congruent to 2 (mod 4).
- N has no prime factors congruent to [7, 19, 41, 53, 61, 71, 73, 77, 83, 89, 91, 97, 103, 107, 109, 119, 127, 139, 161, 173, 181, 191, 193, 197, 203, 209, 211, 217, 223, 227, 229, 239, 247, 259, 281, 293, 301, 311, 313, 317, 323, 329, 331, 337, 343, 347, 349, 359, 367, 379, 401, 413, 421, 431, 433, 437, 443, 449, 451, 457, 463, 467, 469, 479] mod 480 to an odd power.
- If N is odd, then the sum of exponents of the prime factors of N of the classes A,B,C,D,E,F,G are all simultaneously odd, or the sum of exponents of the prime factors of N of the classes A,B,C,D,E,F,G are all simultaneously even, or the sum of exponents of the prime factors of N of these following classes {A,B,C,D or A,C,E or B,D,E or A,B,F or C,D,F or B,C,E,F or A,D,E,F or B,C,G or A,D,G or A,B,E,G or C,D,E,G or A,C,F,G or B,D,F,G or E,F,G} are all simultaneously odd, with the sum of exponents of the prime factors of N of the rest of the classes are all simultaneously even.
- If the highest power of 2 dividing N is an even number that is two or more, then the sum of exponents of the prime factors of N of the classes A,B,C,D,E,F,G are all simultaneously odd, or the sum of exponents of the prime factors of N of the classes A,B,C,D,E,F,G are all simultaneously even, or the sum of exponents of the prime factors of N of these following classes {B,C or A,D or A,B,C,D or A,B,E or A,C,E or B,D,E or C,D,E or A,B,F or A,C,F or B,D,F or C,D,F or E,F or B,C,E,F or A,D,E,F or A,B,C,D,E,F or G or B,C,G or A,D,G or A,B,C,D,G or A,B,E,G or A,C,E,G or B,D,E,G or C,D,E,G or A,B,F,G or A,C,F,G or B,D,F,G or C,D,F,G or E,F,G or B,C,E,F,G or A,D,E,F,G} are all simultaneously odd, with the sum of exponents of the prime factors of N of the rest of the classes are all simultaneously even.
- If the highest power of 2 dividing N is an odd number that is three or more, then the sum of exponents of the prime factors of N of these following classes {A,B or A,C or B,D or C,D or E or B,C,E or A,D,E or A,B,C,D,E or F or B,C,F or A,D,F or A,B,C,D,F or A,B,E,F or A,C,E,F or B,D,E,F or C,D,E,F or A,B,G or A,C,G or B,D,G or C,D,G or E,G or B,C,E,G or A,D,E,G or A,B,C,D,E,G or F,G or B,C,F,G or A,D,F,G or A,B,C,D,F,G or A,B,E,F,G or A,C,E,F,G or B,D,E,F,G or C,D,E,F,G} are all simultaneously odd, with the sum of exponents of the prime factors of N of the rest of the classes are all simultaneously even.

[Raman]

k = 165

Primes p of the form a²+165b²: All primes congruent to [1, 49, 169, 181, 229, 289, 301, 361, 421, 529] mod 660.
if N is a non-negative integer that can be written as a²+165b², then p × N can be written as a²+165b²
if N is a non-negative integer that cannot be written as a²+165b², then p × N cannot be written as a²+165b²

Class A primes: All primes congruent to [2, 83, 107, 167, 227, 263, 347, 503, 527, 563, 623] mod 660
Class B primes: All primes congruent to [3, 67, 103, 163, 223, 247, 367, 427, 463, 487, 643] mod 660
Class C primes: All primes congruent to [5, 53, 113, 137, 257, 317, 353, 377, 533, 617, 653] mod 660
Class D primes: All primes congruent to [11, 59, 71, 119, 179, 191, 251, 311, 419, 551, 599] mod 660
Class E primes: All primes congruent to [13, 73, 193, 217, 277, 337, 373, 457, 613, 637] mod 660
Class F primes: All primes congruent to [19, 79, 139, 151, 211, 259, 271, 391, 439, 571] mod 660
Class G primes: All primes congruent to [29, 41, 101, 149, 161, 281, 329, 461, 569, 629] mod 660

N can be written as a²+165b² if and only if
- N has no prime factors congruent to [7, 17, 23, 31, 37, 43, 47, 61, 89, 91, 97, 109, 127, 131, 133, 157, 173, 197, 199, 203, 221, 233, 239, 241, 269, 283, 287, 293, 299, 307, 313, 323, 331, 343, 349, 359, 371, 379, 383, 389, 397, 401, 403, 409, 413, 431, 433, 437, 443, 449, 467, 469, 479, 481, 491, 493, 497, 499, 509, 511, 521, 523, 541, 547, 553, 557, 559, 577, 581, 587, 589, 593, 601, 607, 611, 619, 631, 641, 647, 659] mod 660 to an odd power.
- Sum of exponents of the prime factors of N of the classes A,B,C,D,E,F,G are all simultaneously odd, or the sum of exponents of the prime factors of N of the classes A,B,C,D,E,F,G are all simultaneously even, or the sum of exponents of the prime factors of N of these following classes {B,C,D or A,B,C,E or A,D,E or A,C,F or A,B,D,F or B,E,F or C,D,E,F or A,B,G or A,C,D,G or C,E,G or B,D,E,G or B,C,F,G or D,F,G or A,E,F,G} are all simultaneously odd, with the sum of exponents of the prime factors of N of the rest of the classes are all simultaneously even.

k = 168

Primes p of the form a²+168b²: All primes congruent to [1, 25, 121, 169, 193, 289, 337, 361, 457, 505, 529, 625] mod 672.
if N is a non-negative integer that can be written as a²+168b², then p × N can be written as a²+168b²
if N is a non-negative integer that cannot be written as a²+168b², then p × N cannot be written as a²+168b²

Class A primes: All primes congruent to [3, 59, 83, 131, 227, 251, 299, 395, 419, 467, 563, 587, 635] mod 672
Class B primes: All primes congruent to [7, 31, 55, 103, 199, 223, 271, 367, 391, 439, 535, 559, 607] mod 672
Class C primes: All primes congruent to [13, 61, 157, 181, 229, 325, 349, 397, 493, 517, 565, 661] mod 672
Class D primes: All primes congruent to [17, 41, 89, 185, 209, 257, 353, 377, 425, 521, 545, 593] mod 672
Class E primes: All primes congruent to [23, 71, 95, 191, 239, 263, 359, 407, 431, 527, 575, 599] mod 672
Class F primes: All primes congruent to [29, 53, 149, 197, 221, 317, 365, 389, 485, 533, 557, 653] mod 672
Class G primes: All primes congruent to [43, 67, 163, 211, 235, 331, 379, 403, 499, 547, 571, 667] mod 672

N can be written as a²+168b² if and only if
- N is not congruent to 2 (mod 4).
- N has no prime factors congruent to [5, 11, 19, 37, 47, 65, 73, 79, 85, 97, 101, 107, 109, 113, 115, 125, 127, 137, 139, 143, 145, 151, 155, 167, 173, 179, 187, 205, 215, 233, 241, 247, 253, 265, 269, 275, 277, 281, 283, 293, 295, 305, 307, 311, 313, 319, 323, 335, 341, 347, 355, 373, 383, 401, 409, 415, 421, 433, 437, 443, 445, 449, 451, 461, 463, 473, 475, 479, 481, 487, 491, 503, 509, 515, 523, 541, 551, 569, 577, 583, 589, 601, 605, 611, 613, 617, 619, 629, 631, 641, 643, 647, 649, 655, 659, 671] mod 672 to an odd power.
- If N is odd, then the sum of exponents of the prime factors of N of the classes A,B,C,D,E,F,G are all simultaneously odd, or the sum of exponents of the prime factors of N of the classes A,B,C,D,E,F,G are all simultaneously even, or the sum of exponents of the prime factors of N of these following classes {A,B,C,D or A,C,E or B,D,E or A,B,F or C,D,F or B,C,E,F or A,D,E,F or B,C,G or A,D,G or A,B,E,G or C,D,E,G or A,C,F,G or B,D,F,G or E,F,G} are all simultaneously odd, with the sum of exponents of the prime factors of N of the rest of the classes are all simultaneously even.
- If the highest power of 2 dividing N is an even number that is two or more, then the sum of exponents of the prime factors of N of the classes A,B,C,D,E,F,G are all simultaneously odd, or the sum of exponents of the prime factors of N of the classes A,B,C,D,E,F,G are all simultaneously even, or the sum of exponents of the prime factors of N of these following classes {B,C or A,D or A,B,C,D or A,B,E or A,C,E or B,D,E or C,D,E or A,B,F or A,C,F or B,D,F or C,D,F or E,F or B,C,E,F or A,D,E,F or A,B,C,D,E,F or G or B,C,G or A,D,G or A,B,C,D,G or A,B,E,G or A,C,E,G or B,D,E,G or C,D,E,G or A,B,F,G or A,C,F,G or B,D,F,G or C,D,F,G or E,F,G or B,C,E,F,G or A,D,E,F,G} are all simultaneously odd, with the sum of exponents of the prime factors of N of the rest of the classes are all simultaneously even.
- If the highest power of 2 dividing N is an odd number that is three or more, then the sum of exponents of the prime factors of N of these following classes {A,B or A,C or B,D or C,D or E or B,C,E or A,D,E or A,B,C,D,E or F or B,C,F or A,D,F or A,B,C,D,F or A,B,E,F or A,C,E,F or B,D,E,F or C,D,E,F or A,B,G or A,C,G or B,D,G or C,D,G or E,G or B,C,E,G or A,D,E,G or A,B,C,D,E,G or F,G or B,C,F,G or A,D,F,G or A,B,C,D,F,G or A,B,E,F,G or A,C,E,F,G or B,D,E,F,G or C,D,E,F,G} are all simultaneously odd, with the sum of exponents of the prime factors of N of the rest of the classes are all simultaneously even.

k = 210

Primes p of the form a²+210b²: All primes congruent to [1, 121, 169, 211, 289, 331, 361, 379, 499, 529, 571, 739] mod 840.
if N is a non-negative integer that can be written as a²+210b², then p × N can be written as a²+210b²
if N is a non-negative integer that cannot be written as a²+210b², then p × N cannot be written as a²+210b²

Class A primes: All primes congruent to [2, 107, 113, 137, 233, 323, 347, 443, 473, 617, 683, 737, 827] mod 840
Class B primes: All primes congruent to [3, 73, 97, 187, 283, 307, 313, 433, 523, 577, 643, 787, 817] mod 840
Class C primes: All primes congruent to [5, 47, 143, 167, 173, 293, 383, 437, 503, 647, 677, 773, 797] mod 840
Class D primes: All primes congruent to [7, 37, 127, 247, 253, 277, 373, 463, 487, 583, 613, 757, 823] mod 840
Class E primes: All primes congruent to [29, 71, 149, 191, 221, 239, 359, 389, 431, 599, 701, 821] mod 840
Class F primes: All primes congruent to [31, 61, 181, 199, 229, 271, 349, 391, 439, 559, 661, 829] mod 840
Class G primes: All primes congruent to [41, 59, 89, 131, 209, 251, 299, 419, 521, 689, 731, 761] mod 840

N can be written as a²+210b² if and only if
- N has no prime factors congruent to [11, 13, 17, 19, 23, 43, 53, 67, 79, 83, 101, 103, 109, 139, 151, 157, 163, 179, 193, 197, 223, 227, 241, 257, 263, 269, 281, 311, 317, 319, 337, 341, 353, 367, 377, 397, 401, 403, 407, 409, 421, 449, 451, 457, 461, 467, 479, 481, 491, 493, 509, 517, 527, 533, 541, 547, 551, 557, 563, 569, 587, 589, 593, 601, 607, 611, 619, 629, 631, 641, 649, 653, 659, 667, 671, 673, 691, 697, 703, 709, 713, 719, 727, 733, 743, 751, 767, 769, 779, 781, 793, 799, 803, 809, 811, 839] mod 840 to an odd power.
- Sum of exponents of the prime factors of N of the classes A,B,C,D,E,F,G are all simultaneously odd, or the sum of exponents of the prime factors of N of the classes A,B,C,D,E,F,G are all simultaneously even, or the sum of exponents of the prime factors of N of these following classes {A,B,C,D or B,C,E or A,D,E or A,C,F or B,D,F or A,B,E,F or C,D,E,F or A,B,G or C,D,G or A,C,E,G or B,D,E,G or B,C,F,G or A,D,F,G or E,F,G} are all simultaneously odd, with the sum of exponents of the prime factors of N of the rest of the classes are all simultaneously even.

[Raman]

k = 240

Primes p of the form a²+240b²: All primes congruent to [1, 49, 121, 169, 241, 289, 361, 409, 481, 529, 601, 649, 721, 769, 841, 889] mod 960.
if N is a non-negative integer that can be written as a²+240b², then p × N can be written as a²+240b²
if N is a non-negative integer that cannot be written as a²+240b², then p × N cannot be written as a²+240b²

Class A primes: All primes congruent to [3, 83, 107, 203, 227, 323, 347, 443, 467, 563, 587, 683, 707, 803, 827, 923, 947] mod 960
Class B primes: All primes congruent to [5, 53, 77, 173, 197, 293, 317, 413, 437, 533, 557, 653, 677, 773, 797, 893, 917] mod 960
Class C primes: All primes congruent to [17, 113, 137, 233, 257, 353, 377, 473, 497, 593, 617, 713, 737, 833, 857, 953] mod 960
Class D primes: All primes congruent to [19, 91, 139, 211, 259, 331, 379, 451, 499, 571, 619, 691, 739, 811, 859, 931] mod 960
Class E primes: All primes congruent to [23, 47, 143, 167, 263, 287, 383, 407, 503, 527, 623, 647, 743, 767, 863, 887] mod 960
Class F primes: All primes congruent to [31, 79, 151, 199, 271, 319, 391, 439, 511, 559, 631, 679, 751, 799, 871, 919] mod 960
Class G primes: All primes congruent to [61, 109, 181, 229, 301, 349, 421, 469, 541, 589, 661, 709, 781, 829, 901, 949] mod 960

N can be written as a²+240b² if and only if
- N is not congruent to 2 (mod 4) or 8 (mod 16) or 32 (mod 64).
- N has no prime factors congruent to [7, 11, 13, 29, 37, 41, 43, 59, 67, 71, 73, 89, 97, 101, 103, 119, 127, 131, 133, 149, 157, 161, 163, 179, 187, 191, 193, 209, 217, 221, 223, 239, 247, 251, 253, 269, 277, 281, 283, 299, 307, 311, 313, 329, 337, 341, 343, 359, 367, 371, 373, 389, 397, 401, 403, 419, 427, 431, 433, 449, 457, 461, 463, 479, 487, 491, 493, 509, 517, 521, 523, 539, 547, 551, 553, 569, 577, 581, 583, 599, 607, 611, 613, 629, 637, 641, 643, 659, 667, 671, 673, 689, 697, 701, 703, 719, 727, 731, 733, 749, 757, 761, 763, 779, 787, 791, 793, 809, 817, 821, 823, 839, 847, 851, 853, 869, 877, 881, 883, 899, 907, 911, 913, 929, 937, 941, 943, 959] mod 960 to an odd power.
- If N is odd, then the sum of exponents of the prime factors of N of the classes A,B,C,D,E,F,G are all simultaneously odd, or the sum of exponents of the prime factors of N of the classes A,B,C,D,E,F,G are all simultaneously even, or the sum of exponents of the prime factors of N of these following classes {A,C,D or A,B,C,E or B,D,E or A,B,F or B,C,D,F or C,E,F or A,D,E,F or B,C,G or A,B,D,G or A,E,G or C,D,E,G or A,C,F,G or D,F,G or B,E,F,G} are all simultaneously odd, with the sum of exponents of the prime factors of N of the rest of the classes are all simultaneously even.
- If N is congruent to 4 (mod 8), then the sum of exponents of the prime factors of N of the classes A,B,C,D,E,F,G are all simultaneously odd, or the sum of exponents of the prime factors of N of the classes A,B,C,D,E,F,G are all simultaneously even, or the sum of exponents of the prime factors of N of these following classes {B,C or A,B,D or A,C,D or A,E or A,B,C,E or B,D,E or C,D,E or A,B,F or A,C,F or D,F or B,C,D,F or B,E,F or C,E,F or A,D,E,F or A,B,C,D,E,F or G or B,C,G or A,B,D,G or A,C,D,G or A,E,G or A,B,C,E,G or B,D,E,G or C,D,E,G or A,B,F,G or A,C,F,G or D,F,G or B,C,D,F,G or B,E,F,G or C,E,F,G or A,D,E,F,G} are all simultaneously odd, with the sum of exponents of the prime factors of N of the rest of the classes are all simultaneously even.
- If the highest power of 2 dividing N is an even number that is four or more, then the sum of exponents of the prime factors of N of the classes A,B,C,D,E,F,G are all simultaneously odd, or the sum of exponents of the prime factors of N of the classes A,B,C,D,E,F,G are all simultaneously even, or the sum of exponents of the prime factors of N of these following classes {A,B or A,C or B,C or D or A,B,D or A,C,D or B,C,D or A,E or B,E or C,E or A,B,C,E or A,D,E or B,D,E or C,D,E or A,B,C,D,E or F or A,B,F or A,C,F or B,C,F or D,F or A,B,D,F or A,C,D,F or B,C,D,F or A,E,F or B,E,F or C,E,F or A,B,C,E,F or A,D,E,F or B,D,E,F or C,D,E,F or A,B,C,D,E,F or G or A,B,G or A,C,G or B,C,G or D,G or A,B,D,G or A,C,D,G or B,C,D,G or A,E,G or B,E,G or C,E,G or A,B,C,E,G or A,D,E,G or B,D,E,G or C,D,E,G or A,B,C,D,E,G or F,G or A,B,F,G or A,C,F,G or B,C,F,G or D,F,G or A,B,D,F,G or A,C,D,F,G or B,C,D,F,G or A,E,F,G or B,E,F,G or C,E,F,G or A,B,C,E,F,G or A,D,E,F,G or B,D,E,F,G or C,D,E,F,G} are all simultaneously odd, with the sum of exponents of the prime factors of N of the rest of the classes are all simultaneously even.
- If the highest power of 2 dividing N is an odd number that is seven or more, then the sum of exponents of the prime factors of N of these following classes {A or B or C or A,B,C or A,D or B,D or C,D or A,B,C,D or E or A,B,E or A,C,E or B,C,E or D,E or A,B,D,E or A,C,D,E or B,C,D,E or A,F or B,F or C,F or A,B,C,F or A,D,F or B,D,F or C,D,F or A,B,C,D,F or E,F or A,B,E,F or A,C,E,F or B,C,E,F or D,E,F or A,B,D,E,F or A,C,D,E,F or B,C,D,E,F or A,G or B,G or C,G or A,B,C,G or A,D,G or B,D,G or C,D,G or A,B,C,D,G or E,G or A,B,E,G or A,C,E,G or B,C,E,G or D,E,G or A,B,D,E,G or A,C,D,E,G or B,C,D,E,G or A,F,G or B,F,G or C,F,G or A,B,C,F,G or A,D,F,G or B,D,F,G or C,D,F,G or A,B,C,D,F,G or E,F,G or A,B,E,F,G or A,C,E,F,G or B,C,E,F,G or D,E,F,G or A,B,D,E,F,G or A,C,D,E,F,G or B,C,D,E,F,G} are all simultaneously odd, with the sum of exponents of the prime factors of N of the rest of the classes are all simultaneously even.

k = 273

Primes p of the form a²+273b²: All primes congruent to [1, 25, 121, 205, 277, 289, 337, 361, 373, 445, 529, 589, 625, 673, 757, 781, 841, 961] mod 1092.
if N is a non-negative integer that can be written as a²+273b², then p × N can be written as a²+273b²
if N is a non-negative integer that cannot be written as a²+273b², then p × N cannot be written as a²+273b²

Class A primes: All primes congruent to [2, 137, 149, 197, 281, 305, 317, 401, 449, 473, 557, 617, 785, 821, 869, 905, 977, 1061, 1073] mod 1092
Class B primes: All primes congruent to [3, 55, 103, 139, 199, 283, 355, 367, 391, 439, 451, 523, 607, 703, 727, 859, 979, 1039, 1063] mod 1092
Class C primes: All primes congruent to [7, 67, 151, 163, 319, 331, 379, 463, 487, 499, 583, 631, 655, 739, 799, 967, 1003, 1051, 1087] mod 1092
Class D primes: All primes congruent to [13, 73, 97, 145, 229, 241, 265, 349, 397, 409, 565, 577, 661, 733, 769, 817, 853, 1021, 1081] mod 1092
Class E primes: All primes congruent to [17, 101, 173, 185, 209, 257, 269, 341, 425, 521, 545, 677, 797, 857, 881, 965, 1013, 1049] mod 1092
Class F primes: All primes congruent to [23, 95, 107, 155, 179, 191, 263, 347, 407, 443, 491, 575, 599, 659, 779, 911, 935, 1031] mod 1092
Class G primes: All primes congruent to [47, 59, 83, 167, 215, 227, 383, 395, 479, 551, 587, 635, 671, 839, 899, 983, 1007, 1055] mod 1092

N can be written as a²+273b² if and only if
- N has no prime factors congruent to [5, 11, 19, 29, 31, 37, 41, 43, 53, 61, 71, 79, 85, 89, 109, 113, 115, 125, 127, 131, 157, 181, 187, 193, 211, 223, 233, 235, 239, 251, 253, 271, 275, 293, 295, 307, 311, 313, 323, 335, 353, 359, 365, 389, 415, 419, 421, 431, 433, 437, 457, 461, 467, 475, 485, 493, 503, 505, 509, 515, 517, 527, 535, 541, 547, 563, 569, 571, 593, 601, 605, 613, 619, 629, 641, 643, 647, 649, 653, 667, 683, 685, 691, 695, 697, 701, 709, 713, 719, 725, 731, 737, 743, 745, 751, 755, 761, 773, 775, 787, 803, 809, 811, 815, 823, 827, 829, 835, 851, 863, 865, 877, 883, 887, 893, 895, 901, 907, 913, 919, 925, 929, 937, 941, 943, 947, 953, 955, 971, 985, 989, 991, 995, 997, 1009, 1019, 1025, 1033, 1037, 1045, 1067, 1069, 1075, 1091] mod 1092 to an odd power.
- Sum of exponents of the prime factors of N of the classes A,B,C,D,E,F,G are all simultaneously odd, or the sum of exponents of the prime factors of N of the classes A,B,C,D,E,F,G are all simultaneously even, or the sum of exponents of the prime factors of N of these following classes {B,C,D or A,B,C,E or A,D,E or A,C,F or A,B,D,F or B,E,F or C,D,E,F or A,B,G or A,C,D,G or C,E,G or B,D,E,G or B,C,F,G or D,F,G or A,E,F,G} are all simultaneously odd, with the sum of exponents of the prime factors of N of the rest of the classes are all simultaneously even.

[Raman]

k = 280

Primes p of the form a²+280b²: All primes congruent to [1, 9, 81, 121, 169, 249, 281, 289, 361, 401, 449, 529, 561, 569, 641, 681, 729, 809, 841, 849, 921, 961, 1009, 1089] mod 1120.
if N is a non-negative integer that can be written as a²+280b², then p × N can be written as a²+280b²
if N is a non-negative integer that cannot be written as a²+280b², then p × N cannot be written as a²+280b²

Class A primes: All primes congruent to [5, 61, 69, 101, 181, 229, 269, 341, 349, 381, 461, 509, 549, 621, 629, 661, 741, 789, 829, 901, 909, 941, 1021, 1069, 1109] mod 1120
Class B primes: All primes congruent to [7, 47, 87, 103, 143, 167, 223, 327, 367, 383, 423, 447, 503, 607, 647, 663, 703, 727, 783, 887, 927, 943, 983, 1007, 1063] mod 1120
Class C primes: All primes congruent to [17, 33, 73, 97, 153, 257, 297, 313, 353, 377, 433, 537, 577, 593, 633, 657, 713, 817, 857, 873, 913, 937, 993, 1097] mod 1120
Class D primes: All primes congruent to [19, 59, 131, 139, 171, 251, 299, 339, 411, 419, 451, 531, 579, 619, 691, 699, 731, 811, 859, 899, 971, 979, 1011, 1091] mod 1120
Class E primes: All primes congruent to [37, 53, 93, 197, 253, 277, 317, 333, 373, 477, 533, 557, 597, 613, 653, 757, 813, 837, 877, 893, 933, 1037, 1093, 1117] mod 1120
Class F primes: All primes congruent to [43, 67, 107, 123, 163, 267, 323, 347, 387, 403, 443, 547, 603, 627, 667, 683, 723, 827, 883, 907, 947, 963, 1003, 1107] mod 1120
Class G primes: All primes congruent to [39, 71, 79, 151, 191, 239, 319, 351, 359, 431, 471, 519, 599, 631, 639, 711, 751, 799, 879, 911, 919, 991, 1031, 1079] mod 1120

N can be written as a²+280b² if and only if
- N is not congruent to 2 (mod 4).
- N has no prime factors congruent to [3, 11, 13, 23, 27, 29, 31, 41, 51, 57, 83, 89, 99, 109, 111, 113, 117, 127, 129, 137, 141, 149, 157, 159, 173, 177, 179, 183, 187, 193, 199, 201, 207, 209, 211, 213, 219, 221, 227, 233, 237, 241, 243, 247, 261, 263, 271, 279, 283, 291, 293, 303, 307, 309, 311, 321, 331, 337, 363, 369, 379, 389, 391, 393, 397, 407, 409, 417, 421, 429, 437, 439, 453, 457, 459, 463, 467, 473, 479, 481, 487, 489, 491, 493, 499, 501, 507, 513, 517, 521, 523, 527, 541, 543, 551, 559, 563, 571, 573, 583, 587, 589, 591, 601, 611, 617, 643, 649, 659, 669, 671, 673, 677, 687, 689, 697, 701, 709, 717, 719, 733, 737, 739, 743, 747, 753, 759, 761, 767, 769, 771, 773, 779, 781, 787, 793, 797, 801, 803, 807, 821, 823, 831, 839, 843, 851, 853, 863, 867, 869, 871, 881, 891, 897, 923, 929, 939, 949, 951, 953, 957, 967, 969, 977, 981, 989, 997, 999, 1013, 1017, 1019, 1023, 1027, 1033, 1039, 1041, 1047, 1049, 1051, 1053, 1059, 1061, 1067, 1073, 1077, 1081, 1083, 1087, 1101, 1103, 1111, 1119] mod 1120 to an odd power.
- If N is odd, then the sum of exponents of the prime factors of N of the classes A,B,C,D,E,F,G are all simultaneously odd, or the sum of exponents of the prime factors of N of the classes A,B,C,D,E,F,G are all simultaneously even, or the sum of exponents of the prime factors of N of these following classes {A,B,C,D or A,C,E or B,D,E or A,B,F or C,D,F or B,C,E,F or A,D,E,F or B,C,G or A,D,G or A,B,E,G or C,D,E,G or A,C,F,G or B,D,F,G or E,F,G} are all simultaneously odd, with the sum of exponents of the prime factors of N of the rest of the classes are all simultaneously even.
- If the highest power of 2 dividing N is an even number that is two or more, then the sum of exponents of the prime factors of N of the classes A,B,C,D,E,F,G are all simultaneously odd, or the sum of exponents of the prime factors of N of the classes A,B,C,D,E,F,G are all simultaneously even, or the sum of exponents of the prime factors of N of these following classes {B,C or A,D or A,B,C,D or A,B,E or A,C,E or B,D,E or C,D,E or A,B,F or A,C,F or B,D,F or C,D,F or E,F or B,C,E,F or A,D,E,F or A,B,C,D,E,F or G or B,C,G or A,D,G or A,B,C,D,G or A,B,E,G or A,C,E,G or B,D,E,G or C,D,E,G or A,B,F,G or A,C,F,G or B,D,F,G or C,D,F,G or E,F,G or B,C,E,F,G or A,D,E,F,G} are all simultaneously odd, with the sum of exponents of the prime factors of N of the rest of the classes are all simultaneously even.
- If the highest power of 2 dividing N is an odd number that is three or more, then the sum of exponents of the prime factors of N of these following classes {A,B or A,C or B,D or C,D or E or B,C,E or A,D,E or A,B,C,D,E or F or B,C,F or A,D,F or A,B,C,D,F or A,B,E,F or A,C,E,F or B,D,E,F or C,D,E,F or A,B,G or A,C,G or B,D,G or C,D,G or E,G or B,C,E,G or A,D,E,G or A,B,C,D,E,G or F,G or B,C,F,G or A,D,F,G or A,B,C,D,F,G or A,B,E,F,G or A,C,E,F,G or B,D,E,F,G or C,D,E,F,G} are all simultaneously odd, with the sum of exponents of the prime factors of N of the rest of the classes are all simultaneously even.

k = 312

Primes p of the form a²+312b²: All primes congruent to [1, 25, 49, 121, 217, 289, 313, 337, 361, 433, 529, 601, 625, 649, 673, 745, 841, 913, 937, 961, 985, 1057, 1153, 1225] mod 1248.
if N is a non-negative integer that can be written as a²+312b², then p × N can be written as a²+312b²
if N is a non-negative integer that cannot be written as a²+312b², then p × N cannot be written as a²+312b²

Class A primes: All primes congruent to [3, 35, 107, 131, 155, 179, 251, 347, 419, 443, 467, 491, 563, 659, 731, 755, 779, 803, 875, 971, 1043, 1067, 1091, 1115, 1187] mod 1248
Class B primes: All primes congruent to [13, 37, 85, 109, 229, 253, 301, 349, 397, 421, 541, 565, 613, 661, 709, 733, 853, 877, 925, 973, 1021, 1045, 1165, 1189, 1237] mod 1248
Class C primes: All primes congruent to [19, 67, 115, 163, 187, 307, 331, 379, 427, 475, 499, 619, 643, 691, 739, 787, 811, 931, 955, 1003, 1051, 1099, 1123, 1243] mod 1248
Class D primes: All primes congruent to [29, 53, 77, 101, 173, 269, 341, 365, 389, 413, 485, 581, 653, 677, 701, 725, 797, 893, 965, 989, 1013, 1037, 1109, 1205] mod 1248
Class E primes: All primes congruent to [41, 89, 137, 161, 281, 305, 353, 401, 449, 473, 593, 617, 665, 713, 761, 785, 905, 929, 977, 1025, 1073, 1097, 1217, 1241] mod 1248
Class F primes: All primes congruent to [47, 71, 119, 167, 215, 239, 359, 383, 431, 479, 527, 551, 671, 695, 743, 791, 839, 863, 983, 1007, 1055, 1103, 1151, 1175] mod 1248
Class G primes: All primes congruent to [55, 79, 103, 127, 199, 295, 367, 391, 415, 439, 511, 607, 679, 703, 727, 751, 823, 919, 991, 1015, 1039, 1063, 1135, 1231] mod 1248

N can be written as a²+312b² if and only if
- N is not congruent to 2 (mod 4).
- N has no prime factors congruent to [5, 7, 11, 17, 23, 31, 43, 59, 61, 73, 83, 95, 97, 113, 125, 133, 139, 145, 149, 151, 157, 175, 181, 185, 191, 193, 197, 203, 205, 209, 211, 223, 227, 233, 235, 241, 245, 257, 259, 263, 265, 271, 275, 277, 283, 287, 293, 311, 317, 319, 323, 329, 335, 343, 355, 371, 373, 385, 395, 407, 409, 425, 437, 445, 451, 457, 461, 463, 469, 487, 493, 497, 503, 505, 509, 515, 517, 521, 523, 535, 539, 545, 547, 553, 557, 569, 571, 575, 577, 583, 587, 589, 595, 599, 605, 623, 629, 631, 635, 641, 647, 655, 667, 683, 685, 697, 707, 719, 721, 737, 749, 757, 763, 769, 773, 775, 781, 799, 805, 809, 815, 817, 821, 827, 829, 833, 835, 847, 851, 857, 859, 865, 869, 881, 883, 887, 889, 895, 899, 901, 907, 911, 917, 935, 941, 943, 947, 953, 959, 967, 979, 995, 997, 1009, 1019, 1031, 1033, 1049, 1061, 1069, 1075, 1081, 1085, 1087, 1093, 1111, 1117, 1121, 1127, 1129, 1133, 1139, 1141, 1145, 1147, 1159, 1163, 1169, 1171, 1177, 1181, 1193, 1195, 1199, 1201, 1207, 1211, 1213, 1219, 1223, 1229, 1247] mod 1248 to an odd power.
- If N is odd, then the sum of exponents of the prime factors of N of the classes A,B,C,D,E,F,G are all simultaneously odd, or the sum of exponents of the prime factors of N of the classes A,B,C,D,E,F,G are all simultaneously even, or the sum of exponents of the prime factors of N of these following classes {A,B,C,D or A,C,E or B,D,E or A,B,F or C,D,F or B,C,E,F or A,D,E,F or B,C,G or A,D,G or A,B,E,G or C,D,E,G or A,C,F,G or B,D,F,G or E,F,G} are all simultaneously odd, with the sum of exponents of the prime factors of N of the rest of the classes are all simultaneously even.
- If the highest power of 2 dividing N is an even number that is two or more, then the sum of exponents of the prime factors of N of the classes A,B,C,D,E,F,G are all simultaneously odd, or the sum of exponents of the prime factors of N of the classes A,B,C,D,E,F,G are all simultaneously even, or the sum of exponents of the prime factors of N of these following classes {B,C or A,D or A,B,C,D or A,B,E or A,C,E or B,D,E or C,D,E or A,B,F or A,C,F or B,D,F or C,D,F or E,F or B,C,E,F or A,D,E,F or A,B,C,D,E,F or G or B,C,G or A,D,G or A,B,C,D,G or A,B,E,G or A,C,E,G or B,D,E,G or C,D,E,G or A,B,F,G or A,C,F,G or B,D,F,G or C,D,F,G or E,F,G or B,C,E,F,G or A,D,E,F,G} are all simultaneously odd, with the sum of exponents of the prime factors of N of the rest of the classes are all simultaneously even.
- If the highest power of 2 dividing N is an odd number that is three or more, then the sum of exponents of the prime factors of N of these following classes {A,B or A,C or B,D or C,D or E or B,C,E or A,D,E or A,B,C,D,E or F or B,C,F or A,D,F or A,B,C,D,F or A,B,E,F or A,C,E,F or B,D,E,F or C,D,E,F or A,B,G or A,C,G or B,D,G or C,D,G or E,G or B,C,E,G or A,D,E,G or A,B,C,D,E,G or F,G or B,C,F,G or A,D,F,G or A,B,C,D,F,G or A,B,E,F,G or A,C,E,F,G or B,D,E,F,G or C,D,E,F,G} are all simultaneously odd, with the sum of exponents of the prime factors of N of the rest of the classes are all simultaneously even.

[Raman]

k = 330

Primes p of the form a²+330b²: All primes congruent to [1, 49, 91, 169, 289, 331, 361, 379, 499, 529, 619, 691, 841, 859, 889, 961, 1081, 1171, 1219, 1291] mod 1320.
if N is a non-negative integer that can be written as a²+330b², then p × N can be written as a²+330b²
if N is a non-negative integer that cannot be written as a²+330b², then p × N cannot be written as a²+330b²

Class A primes: All primes congruent to [2, 167, 173, 197, 263, 293, 413, 437, 503, 527, 557, 623, 677, 743, 767, 887, 893, 1007, 1157, 1223, 1253] mod 1320
Class B primes: All primes congruent to [3, 113, 137, 203, 257, 323, 353, 377, 443, 467, 587, 617, 683, 707, 713, 947, 977, 1043, 1193, 1307, 1313] mod 1320
Class C primes: All primes congruent to [5, 71, 119, 191, 221, 269, 311, 389, 509, 551, 581, 599, 719, 749, 839, 911, 1061, 1079, 1109, 1181, 1301] mod 1320
Class D primes: All primes congruent to [11, 41, 131, 161, 281, 299, 329, 371, 491, 569, 611, 659, 689, 761, 809, 899, 1019, 1091, 1121, 1139, 1289] mod 1320
Class E primes: All primes congruent to [37, 103, 133, 157, 223, 247, 367, 397, 463, 487, 493, 727, 757, 823, 973, 1087, 1093, 1213, 1237, 1303] mod 1320
Class F primes: All primes congruent to [43, 73, 193, 217, 283, 307, 337, 403, 457, 523, 547, 667, 673, 787, 937, 1003, 1033, 1267, 1273, 1297] mod 1320
Class G primes: All primes congruent to [61, 79, 109, 151, 271, 349, 391, 439, 469, 541, 589, 679, 799, 871, 901, 919, 1069, 1141, 1231, 1261] mod 1320

N can be written as a²+330b² if and only if
- N has no prime factors congruent to [7, 13, 17, 19, 23, 29, 31, 47, 53, 59, 67, 83, 89, 97, 101, 107, 127, 139, 149, 163, 179, 181, 199, 211, 227, 229, 233, 239, 241, 251, 259, 277, 287, 301, 313, 317, 343, 347, 359, 373, 383, 401, 409, 419, 421, 427, 431, 433, 449, 461, 479, 481, 497, 511, 521, 533, 553, 559, 563, 571, 577, 593, 601, 607, 613, 629, 631, 637, 641, 643, 647, 653, 661, 697, 701, 703, 709, 721, 731, 733, 739, 751, 763, 769, 773, 779, 791, 793, 797, 811, 817, 821, 827, 829, 833, 851, 853, 857, 863, 877, 881, 883, 907, 917, 923, 929, 931, 941, 943, 949, 953, 959, 967, 971, 983, 989, 991, 997, 1009, 1013, 1021, 1027, 1031, 1037, 1039, 1049, 1051, 1057, 1063, 1073, 1097, 1099, 1103, 1117, 1123, 1127, 1129, 1147, 1151, 1153, 1159, 1163, 1169, 1183, 1187, 1189, 1201, 1207, 1211, 1217, 1229, 1241, 1247, 1249, 1259, 1271, 1277, 1279, 1283, 1319] mod 1320 to an odd power.
- Sum of exponents of the prime factors of N of the classes A,B,C,D,E,F,G are all simultaneously odd, or the sum of exponents of the prime factors of N of the classes A,B,C,D,E,F,G are all simultaneously even, or the sum of exponents of the prime factors of N of these following classes {A,B,C,D or B,C,E or A,D,E or A,C,F or B,D,F or A,B,E,F or C,D,E,F or A,B,G or C,D,G or A,C,E,G or B,D,E,G or B,C,F,G or A,D,F,G or E,F,G} are all simultaneously odd, with the sum of exponents of the prime factors of N of the rest of the classes are all simultaneously even.

k = 345

Primes p of the form a²+345b²: All primes congruent to [1, 49, 121, 169, 289, 301, 349, 361, 409, 469, 541, 601, 721, 829, 841, 901, 949, 961, 1021, 1129, 1189, 1369] mod 1380.
if N is a non-negative integer that can be written as a²+345b², then p × N can be written as a²+345b²
if N is a non-negative integer that cannot be written as a²+345b², then p × N cannot be written as a²+345b²

Class A primes: All primes congruent to [2, 77, 173, 197, 233, 257, 317, 353, 377, 473, 533, 593, 653, 737, 857, 1013, 1037, 1097, 1133, 1277, 1313, 1337, 1373] mod 1380
Class B primes: All primes congruent to [3, 127, 163, 187, 223, 307, 403, 427, 463, 487, 547, 583, 607, 703, 763, 823, 883, 967, 1087, 1243, 1267, 1327, 1363] mod 1380
Class C primes: All primes congruent to [5, 89, 149, 221, 281, 329, 341, 389, 401, 521, 569, 641, 689, 701, 881, 941, 1049, 1109, 1121, 1169, 1229, 1241, 1349] mod 1380
Class D primes: All primes congruent to [19, 79, 91, 199, 319, 379, 451, 511, 559, 571, 619, 631, 751, 799, 871, 919, 931, 1111, 1171, 1279, 1339, 1351] mod 1380
Class E primes: All primes congruent to [23, 83, 107, 143, 203, 227, 263, 287, 383, 467, 503, 527, 563, 707, 743, 803, 827, 983, 1103, 1187, 1247, 1307, 1367] mod 1380
Class F primes: All primes congruent to [37, 97, 157, 217, 313, 337, 373, 433, 457, 493, 517, 613, 697, 733, 757, 793, 937, 973, 1033, 1057, 1213, 1333] mod 1380
Class G primes: All primes congruent to [59, 71, 119, 131, 179, 239, 311, 371, 491, 599, 611, 671, 719, 731, 791, 899, 959, 1139, 1151, 1199, 1271, 1319] mod 1380

N can be written as a²+345b² if and only if
- N has no prime factors congruent to [7, 11, 13, 17, 29, 31, 41, 43, 47, 53, 61, 67, 73, 101, 103, 109, 113, 133, 137, 139, 151, 167, 181, 191, 193, 209, 211, 229, 241, 247, 251, 259, 269, 271, 277, 283, 293, 323, 331, 343, 347, 359, 367, 397, 407, 413, 419, 421, 431, 439, 443, 449, 461, 479, 481, 497, 499, 509, 523, 539, 551, 553, 557, 577, 581, 587, 589, 617, 623, 629, 637, 643, 647, 649, 659, 661, 673, 677, 679, 683, 691, 709, 727, 739, 749, 761, 767, 769, 773, 779, 781, 787, 797, 809, 811, 817, 821, 833, 839, 847, 853, 859, 863, 869, 877, 887, 889, 893, 907, 911, 913, 917, 923, 929, 947, 953, 971, 977, 979, 991, 997, 1001, 1003, 1007, 1009, 1019, 1027, 1031, 1039, 1043, 1051, 1061, 1063, 1067, 1069, 1073, 1079, 1091, 1093, 1099, 1117, 1123, 1141, 1147, 1153, 1157, 1159, 1163, 1177, 1181, 1183, 1193, 1201, 1207, 1211, 1217, 1223, 1231, 1237, 1249, 1253, 1259, 1261, 1273, 1283, 1289, 1291, 1297, 1301, 1303, 1309, 1321, 1331, 1343, 1361, 1379] mod 1380 to an odd power.
- Sum of exponents of the prime factors of N of the classes A,B,C,D,E,F,G are all simultaneously odd, or the sum of exponents of the prime factors of N of the classes A,B,C,D,E,F,G are all simultaneously even, or the sum of exponents of the prime factors of N of these following classes {A,B,C,D or B,C,E or A,D,E or A,C,F or B,D,F or A,B,E,F or C,D,E,F or A,B,G or C,D,G or A,C,E,G or B,D,E,G or B,C,F,G or A,D,F,G or E,F,G} are all simultaneously odd, with the sum of exponents of the prime factors of N of the rest of the classes are all simultaneously even.

[Raman]

k = 357

Primes p of the form a²+357b²: All primes congruent to [1, 25, 121, 169, 205, 253, 361, 373, 421, 457, 529, 613, 625, 757, 781, 841, 865, 961, 1033, 1045, 1177, 1345, 1369, 1381] mod 1428.
if N is a non-negative integer that can be written as a²+357b², then p × N can be written as a²+357b²
if N is a non-negative integer that cannot be written as a²+357b², then p × N cannot be written as a²+357b²

Class A primes: All primes congruent to [2, 155, 179, 191, 239, 263, 359, 407, 443, 491, 599, 611, 659, 695, 767, 851, 863, 995, 1019, 1079, 1103, 1199, 1271, 1283, 1415] mod 1428
Class B primes: All primes congruent to [3, 131, 143, 167, 215, 227, 299, 311, 335, 419, 479, 503, 551, 635, 719, 755, 839, 887, 923, 983, 1091, 1151, 1235, 1319, 1391] mod 1428
Class C primes: All primes congruent to [7, 79, 163, 211, 235, 295, 379, 403, 415, 487, 499, 547, 571, 583, 751, 823, 907, 991, 1051, 1159, 1219, 1255, 1303, 1387, 1423] mod 1428
Class D primes: All primes congruent to [17, 89, 101, 185, 257, 293, 341, 353, 461, 509, 545, 593, 689, 713, 761, 773, 797, 965, 1097, 1109, 1181, 1277, 1301, 1361, 1385] mod 1428
Class E primes: All primes congruent to [19, 55, 103, 115, 223, 271, 307, 355, 451, 475, 523, 535, 559, 727, 859, 871, 943, 1039, 1063, 1123, 1147, 1279, 1291, 1375] mod 1428
Class F primes: All primes congruent to [29, 65, 113, 197, 233, 317, 401, 449, 473, 533, 617, 641, 653, 725, 737, 785, 809, 821, 989, 1061, 1145, 1229, 1289, 1397] mod 1428
Class G primes: All primes congruent to [61, 73, 97, 181, 241, 265, 313, 397, 481, 517, 601, 649, 685, 745, 853, 913, 997, 1081, 1153, 1321, 1333, 1357, 1405, 1417] mod 1428

N can be written as a²+357b² if and only if
- N has no prime factors congruent to [5, 11, 13, 23, 31, 37, 41, 43, 47, 53, 59, 67, 71, 83, 95, 107, 109, 125, 127, 137, 139, 145, 149, 151, 157, 173, 193, 199, 209, 229, 247, 251, 269, 275, 277, 281, 283, 305, 319, 325, 331, 337, 347, 349, 365, 367, 377, 383, 389, 395, 409, 431, 433, 437, 439, 445, 463, 467, 485, 505, 515, 521, 541, 557, 563, 565, 569, 575, 577, 587, 589, 605, 607, 619, 631, 643, 647, 655, 661, 667, 671, 673, 677, 683, 691, 701, 703, 709, 715, 733, 739, 743, 769, 775, 779, 787, 793, 803, 811, 815, 817, 827, 829, 835, 845, 857, 869, 877, 881, 883, 893, 895, 899, 905, 911, 919, 925, 929, 937, 941, 947, 949, 953, 955, 967, 971, 977, 979, 985, 1007, 1009, 1013, 1021, 1025, 1027, 1031, 1049, 1055, 1067, 1069, 1073, 1075, 1087, 1093, 1111, 1115, 1117, 1121, 1129, 1133, 1135, 1157, 1163, 1165, 1171, 1175, 1187, 1189, 1193, 1195, 1201, 1205, 1213, 1217, 1223, 1231, 1237, 1243, 1247, 1249, 1259, 1261, 1265, 1273, 1285, 1297, 1307, 1313, 1315, 1325, 1327, 1331, 1339, 1349, 1355, 1363, 1367, 1373, 1399, 1403, 1409, 1427] mod 1428 to an odd power.
- Sum of exponents of the prime factors of N of the classes A,B,C,D,E,F,G are all simultaneously odd, or the sum of exponents of the prime factors of N of the classes A,B,C,D,E,F,G are all simultaneously even, or the sum of exponents of the prime factors of N of these following classes {B,C,D or A,B,C,E or A,D,E or A,C,F or A,B,D,F or B,E,F or C,D,E,F or A,B,G or A,C,D,G or C,E,G or B,D,E,G or B,C,F,G or D,F,G or A,E,F,G} are all simultaneously odd, with the sum of exponents of the prime factors of N of the rest of the classes are all simultaneously even.

k = 385

Primes p of the form a²+385b²: All primes congruent to [1, 9, 81, 141, 169, 221, 289, 309, 361, 389, 401, 421, 449, 529, 641, 669, 709, 729, 841, 949, 961, 1061, 1101, 1149, 1241, 1269, 1369, 1401, 1409, 1521] mod 1540.
if N is a non-negative integer that can be written as a²+385b², then p × N can be written as a²+385b²
if N is a non-negative integer that cannot be written as a²+385b², then p × N cannot be written as a²+385b²

Class A primes: All primes congruent to [2, 57, 193, 197, 233, 277, 337, 373, 393, 417, 457, 513, 557, 613, 673, 813, 877, 893, 897, 953, 1033, 1073, 1117, 1157, 1173, 1297, 1317, 1437, 1493, 1513, 1537] mod 1540
Class B primes: All primes congruent to [5, 97, 157, 213, 257, 313, 353, 377, 397, 433, 493, 537, 573, 577, 713, 773, 797, 817, 873, 993, 1013, 1137, 1153, 1193, 1237, 1277, 1357, 1413, 1417, 1433, 1497] mod 1540
Class C primes: All primes congruent to [7, 83, 87, 167, 227, 283, 307, 327, 447, 503, 523, 563, 607, 703, 747, 783, 787, 843, 887, 923, 943, 1007, 1063, 1207, 1223, 1363, 1403, 1427, 1447, 1487, 1503] mod 1540
Class D primes: All primes congruent to [11, 39, 51, 79, 151, 211, 219, 239, 351, 359, 431, 459, 491, 519, 571, 611, 659, 711, 739, 799, 879, 919, 1019, 1031, 1051, 1271, 1311, 1339, 1359, 1451, 1471] mod 1540
Class E primes: All primes congruent to [23, 67, 163, 207, 247, 267, 323, 443, 463, 487, 543, 603, 683, 687, 807, 823, 863, 883, 907, 947, 1087, 1103, 1247, 1303, 1367, 1387, 1423, 1467, 1523, 1527] mod 1540
Class F primes: All primes congruent to [31, 59, 111, 159, 199, 251, 279, 311, 339, 411, 419, 531, 551, 559, 619, 691, 719, 731, 839, 859, 951, 971, 999, 1039, 1259, 1279, 1291, 1391, 1431, 1511] mod 1540
Class G primes: All primes congruent to [41, 61, 101, 129, 241, 321, 349, 369, 381, 409, 461, 481, 549, 601, 629, 689, 761, 769, 789, 901, 909, 941, 1041, 1069, 1161, 1209, 1249, 1349, 1361, 1469] mod 1540

N can be written as a²+385b² if and only if
- N has no prime factors congruent to [3, 13, 17, 19, 27, 29, 37, 43, 47, 53, 69, 71, 73, 89, 93, 103, 107, 109, 113, 117, 123, 127, 131, 137, 139, 149, 153, 171, 173, 177, 179, 181, 183, 191, 201, 223, 229, 237, 243, 249, 261, 263, 269, 271, 281, 291, 293, 299, 303, 317, 331, 333, 347, 367, 379, 383, 387, 391, 403, 423, 437, 439, 453, 467, 471, 477, 479, 489, 499, 501, 507, 509, 521, 527, 533, 541, 547, 569, 579, 587, 589, 591, 593, 597, 599, 617, 621, 631, 633, 639, 643, 647, 653, 657, 661, 663, 667, 677, 681, 697, 699, 701, 717, 723, 727, 733, 741, 743, 751, 753, 757, 767, 771, 779, 793, 801, 809, 811, 821, 827, 829, 831, 837, 849, 851, 853, 857, 867, 871, 881, 899, 911, 921, 927, 929, 933, 937, 939, 963, 967, 969, 977, 981, 983, 989, 991, 997, 1003, 1009, 1011, 1017, 1021, 1027, 1037, 1047, 1049, 1053, 1059, 1077, 1079, 1081, 1083, 1091, 1093, 1097, 1107, 1109, 1119, 1121, 1123, 1129, 1131, 1139, 1143, 1147, 1151, 1159, 1163, 1167, 1171, 1179, 1181, 1187, 1189, 1191, 1201, 1203, 1213, 1217, 1219, 1227, 1229, 1231, 1233, 1251, 1257, 1261, 1263, 1273, 1283, 1289, 1293, 1299, 1301, 1307, 1313, 1319, 1321, 1327, 1329, 1333, 1341, 1343, 1347, 1371, 1373, 1377, 1381, 1383, 1389, 1399, 1411, 1429, 1439, 1443, 1453, 1457, 1459, 1461, 1473, 1479, 1481, 1483, 1489, 1499, 1501, 1509, 1517, 1531, 1539] mod 1540 to an odd power.
- Sum of exponents of the prime factors of N of the classes A,B,C,D,E,F,G are all simultaneously odd, or the sum of exponents of the prime factors of N of the classes A,B,C,D,E,F,G are all simultaneously even, or the sum of exponents of the prime factors of N of these following classes {B,C,D or A,B,C,E or A,D,E or A,C,F or A,B,D,F or B,E,F or C,D,E,F or A,B,G or A,C,D,G or C,E,G or B,D,E,G or B,C,F,G or D,F,G or A,E,F,G} are all simultaneously odd, with the sum of exponents of the prime factors of N of the rest of the classes are all simultaneously even.

[Raman]

k = 408

Primes p of the form a²+408b²: All primes congruent to [1, 25, 49, 121, 145, 169, 217, 361, 409, 433, 457, 529, 553, 577, 625, 769, 817, 841, 865, 937, 961, 985, 1033, 1177, 1225, 1249, 1273, 1345, 1369, 1393, 1441, 1585] mod 1632.
if N is a non-negative integer that can be written as a²+408b², then p × N can be written as a²+408b²
if N is a non-negative integer that cannot be written as a²+408b², then p × N cannot be written as a²+408b²

Class A primes: All primes congruent to [3, 91, 139, 163, 211, 235, 283, 379, 403, 499, 547, 571, 619, 643, 691, 787, 811, 907, 955, 979, 1027, 1051, 1099, 1195, 1219, 1315, 1363, 1387, 1435, 1459, 1507, 1603, 1627] mod 1632
Class B primes: All primes congruent to [17, 41, 65, 113, 209, 233, 329, 377, 401, 449, 473, 521, 617, 641, 737, 785, 809, 857, 881, 929, 1025, 1049, 1145, 1193, 1217, 1265, 1289, 1337, 1433, 1457, 1553, 1601, 1625] mod 1632
Class C primes: All primes congruent to [23, 71, 95, 143, 167, 215, 311, 335, 431, 479, 503, 551, 575, 623, 719, 743, 839, 887, 911, 959, 983, 1031, 1127, 1151, 1247, 1295, 1319, 1367, 1391, 1439, 1535, 1559] mod 1632
Class D primes: All primes congruent to [37, 61, 109, 133, 181, 277, 301, 397, 445, 469, 517, 541, 589, 685, 709, 805, 853, 877, 925, 949, 997, 1093, 1117, 1213, 1261, 1285, 1333, 1357, 1405, 1501, 1525, 1621] mod 1632
Class E primes: All primes congruent to [53, 77, 101, 149, 293, 341, 365, 389, 461, 485, 509, 557, 701, 749, 773, 797, 869, 893, 917, 965, 1109, 1157, 1181, 1205, 1277, 1301, 1325, 1373, 1517, 1565, 1589, 1613] mod 1632
Class F primes: All primes congruent to [35, 59, 83, 155, 179, 203, 251, 395, 443, 467, 491, 563, 587, 611, 659, 803, 851, 875, 899, 971, 995, 1019, 1067, 1211, 1259, 1283, 1307, 1379, 1403, 1427, 1475, 1619] mod 1632
Class G primes: All primes congruent to [55, 103, 127, 151, 223, 247, 271, 319, 463, 511, 535, 559, 631, 655, 679, 727, 871, 919, 943, 967, 1039, 1063, 1087, 1135, 1279, 1327, 1351, 1375, 1447, 1471, 1495, 1543] mod 1632

N can be written as a²+408b² if and only if
- N is not congruent to 2 (mod 4).
- N has no prime factors congruent to [5, 7, 11, 13, 19, 29, 31, 43, 47, 67, 73, 79, 89, 97, 107, 115, 125, 131, 137, 157, 161, 173, 175, 185, 191, 193, 197, 199, 205, 227, 229, 239, 241, 245, 253, 257, 259, 263, 265, 269, 275, 281, 287, 295, 299, 305, 307, 313, 317, 325, 331, 337, 343, 347, 349, 353, 355, 359, 367, 371, 373, 383, 385, 407, 413, 415, 419, 421, 427, 437, 439, 451, 455, 475, 481, 487, 497, 505, 515, 523, 533, 539, 545, 565, 569, 581, 583, 593, 599, 601, 605, 607, 613, 635, 637, 647, 649, 653, 661, 665, 667, 671, 673, 677, 683, 689, 695, 703, 707, 713, 715, 721, 725, 733, 739, 745, 751, 755, 757, 761, 763, 767, 775, 779, 781, 791, 793, 815, 821, 823, 827, 829, 835, 845, 847, 859, 863, 883, 889, 895, 905, 913, 923, 931, 941, 947, 953, 973, 977, 989, 991, 1001, 1007, 1009, 1013, 1015, 1021, 1043, 1045, 1055, 1057, 1061, 1069, 1073, 1075, 1079, 1081, 1085, 1091, 1097, 1103, 1111, 1115, 1121, 1123, 1129, 1133, 1141, 1147, 1153, 1159, 1163, 1165, 1169, 1171, 1175, 1183, 1187, 1189, 1199, 1201, 1223, 1229, 1231, 1235, 1237, 1243, 1253, 1255, 1267, 1271, 1291, 1297, 1303, 1313, 1321, 1331, 1339, 1349, 1355, 1361, 1381, 1385, 1397, 1399, 1409, 1415, 1417, 1421, 1423, 1429, 1451, 1453, 1463, 1465, 1469, 1477, 1481, 1483, 1487, 1489, 1493, 1499, 1505, 1511, 1519, 1523, 1529, 1531, 1537, 1541, 1549, 1555, 1561, 1567, 1571, 1573, 1577, 1579, 1583, 1591, 1595, 1597, 1607, 1609, 1631] mod 1632 to an odd power.
- If N is odd, then the sum of exponents of the prime factors of N of the classes A,B,C,D,E,F,G are all simultaneously odd, or the sum of exponents of the prime factors of N of the classes A,B,C,D,E,F,G are all simultaneously even, or the sum of exponents of the prime factors of N of these following classes {A,B,C,D or A,C,E or B,D,E or A,B,F or C,D,F or B,C,E,F or A,D,E,F or B,C,G or A,D,G or A,B,E,G or C,D,E,G or A,C,F,G or B,D,F,G or E,F,G} are all simultaneously odd, with the sum of exponents of the prime factors of N of the rest of the classes are all simultaneously even.
- If the highest power of 2 dividing N is an even number that is two or more, then the sum of exponents of the prime factors of N of the classes A,B,C,D,E,F,G are all simultaneously odd, or the sum of exponents of the prime factors of N of the classes A,B,C,D,E,F,G are all simultaneously even, or the sum of exponents of the prime factors of N of these following classes {B,C or A,D or A,B,C,D or A,B,E or A,C,E or B,D,E or C,D,E or A,B,F or A,C,F or B,D,F or C,D,F or E,F or B,C,E,F or A,D,E,F or A,B,C,D,E,F or G or B,C,G or A,D,G or A,B,C,D,G or A,B,E,G or A,C,E,G or B,D,E,G or C,D,E,G or A,B,F,G or A,C,F,G or B,D,F,G or C,D,F,G or E,F,G or B,C,E,F,G or A,D,E,F,G} are all simultaneously odd, with the sum of exponents of the prime factors of N of the rest of the classes are all simultaneously even.
- If the highest power of 2 dividing N is an odd number that is three or more, then the sum of exponents of the prime factors of N of these following classes {A,B or A,C or B,D or C,D or E or B,C,E or A,D,E or A,B,C,D,E or F or B,C,F or A,D,F or A,B,C,D,F or A,B,E,F or A,C,E,F or B,D,E,F or C,D,E,F or A,B,G or A,C,G or B,D,G or C,D,G or E,G or B,C,E,G or A,D,E,G or A,B,C,D,E,G or F,G or B,C,F,G or A,D,F,G or A,B,C,D,F,G or A,B,E,F,G or A,C,E,F,G or B,D,E,F,G or C,D,E,F,G} are all simultaneously odd, with the sum of exponents of the prime factors of N of the rest of the classes are all simultaneously even.

k = 462

Primes p of the form a²+462b²: All primes congruent to [1, 25, 169, 247, 289, 295, 361, 463, 487, 529, 625, 631, 697, 751, 793, 823, 841, 961, 991, 1087, 1159, 1255, 1303, 1345, 1369, 1423, 1633, 1681, 1807, 1831] mod 1848.
if N is a non-negative integer that can be written as a²+462b², then p × N can be written as a²+462b²
if N is a non-negative integer that cannot be written as a²+462b², then p × N cannot be written as a²+462b²

Class A primes: All primes congruent to [2, 65, 95, 233, 239, 263, 281, 305, 359, 431, 527, 569, 695, 743, 767, 809, 953, 1031, 1073, 1121, 1271, 1289, 1415, 1481, 1535, 1583, 1625, 1649, 1745, 1751, 1817] mod 1848
Class B primes: All primes congruent to [3, 115, 157, 181, 229, 355, 397, 493, 565, 619, 643, 661, 685, 691, 829, 859, 955, 1021, 1027, 1123, 1147, 1189, 1237, 1291, 1357, 1483, 1501, 1651, 1699, 1741, 1819] mod 1848
Class C primes: All primes congruent to [7, 73, 145, 241, 271, 391, 409, 439, 481, 535, 601, 607, 703, 745, 769, 871, 937, 943, 985, 1063, 1207, 1231, 1249, 1273, 1399, 1447, 1657, 1711, 1735, 1777, 1825] mod 1848
Class D primes: All primes congruent to [11, 53, 155, 179, 221, 317, 323, 389, 443, 485, 515, 533, 653, 683, 779, 851, 947, 995, 1037, 1061, 1115, 1325, 1373, 1499, 1523, 1541, 1565, 1709, 1787, 1829, 1835] mod 1848
Class E primes: All primes congruent to [43, 85, 109, 205, 211, 277, 373, 403, 541, 547, 571, 589, 613, 667, 739, 835, 877, 1003, 1051, 1075, 1117, 1261, 1339, 1381, 1429, 1579, 1597, 1723, 1789, 1843] mod 1848
Class F primes: All primes congruent to [47, 89, 185, 257, 311, 335, 353, 377, 383, 521, 551, 647, 713, 719, 815, 839, 881, 929, 983, 1049, 1175, 1193, 1343, 1391, 1433, 1511, 1655, 1697, 1721, 1769] mod 1848
Class G primes: All primes congruent to [83, 101, 131, 173, 227, 293, 299, 395, 437, 461, 563, 629, 635, 677, 755, 899, 923, 941, 965, 1091, 1139, 1349, 1403, 1427, 1469, 1517, 1613, 1685, 1781, 1811] mod 1848

N can be written as a²+462b² if and only if
- N has no prime factors congruent to [5, 13, 17, 19, 23, 29, 31, 37, 41, 59, 61, 67, 71, 79, 97, 103, 107, 113, 125, 127, 137, 139, 149, 151, 163, 167, 191, 193, 197, 199, 215, 223, 235, 251, 265, 269, 283, 307, 313, 325, 331, 337, 347, 349, 365, 367, 379, 401, 415, 419, 421, 425, 433, 445, 449, 457, 467, 475, 479, 491, 499, 503, 505, 509, 523, 545, 557, 559, 575, 577, 587, 593, 599, 611, 617, 641, 655, 659, 673, 689, 701, 709, 725, 727, 731, 733, 757, 761, 773, 775, 785, 787, 797, 799, 811, 817, 821, 827, 845, 853, 857, 863, 865, 883, 887, 893, 895, 901, 905, 907, 911, 919, 925, 949, 967, 971, 977, 989, 997, 1007, 1009, 1013, 1019, 1025, 1033, 1039, 1055, 1069, 1079, 1081, 1093, 1097, 1103, 1105, 1109, 1129, 1135, 1145, 1151, 1153, 1157, 1163, 1165, 1171, 1181, 1187, 1195, 1201, 1205, 1213, 1217, 1219, 1223, 1229, 1235, 1241, 1247, 1259, 1277, 1279, 1283, 1285, 1297, 1301, 1307, 1313, 1315, 1319, 1321, 1327, 1333, 1355, 1361, 1363, 1367, 1385, 1387, 1405, 1409, 1411, 1417, 1439, 1445, 1451, 1453, 1457, 1459, 1465, 1471, 1475, 1487, 1489, 1493, 1495, 1513, 1525, 1531, 1537, 1543, 1549, 1553, 1555, 1559, 1567, 1571, 1577, 1585, 1591, 1601, 1607, 1609, 1615, 1619, 1621, 1627, 1637, 1643, 1663, 1667, 1669, 1675, 1679, 1691, 1693, 1703, 1717, 1733, 1739, 1747, 1753, 1759, 1763, 1765, 1775, 1783, 1795, 1801, 1805, 1823, 1847] mod 1848 to an odd power.
- Sum of exponents of the prime factors of N of the classes A,B,C,D,E,F,G are all simultaneously odd, or the sum of exponents of the prime factors of N of the classes A,B,C,D,E,F,G are all simultaneously even, or the sum of exponents of the prime factors of N of these following classes {A,B,C,D or B,C,E or A,D,E or A,C,F or B,D,F or A,B,E,F or C,D,E,F or A,B,G or C,D,G or A,C,E,G or B,D,E,G or B,C,F,G or A,D,F,G or E,F,G} are all simultaneously odd, with the sum of exponents of the prime factors of N of the rest of the classes are all simultaneously even.

[Raman]

k = 520

Primes p of the form a²+520b²: All primes congruent to [1, 9, 49, 81, 121, 129, 209, 289, 321, 329, 361, 441, 521, 529, 569, 601, 641, 649, 729, 809, 841, 849, 881, 961, 1041, 1049, 1089, 1121, 1161, 1169, 1249, 1329, 1361, 1369, 1401, 1481, 1561, 1569, 1609, 1641, 1681, 1689, 1769, 1849, 1881, 1889, 1921, 2001] mod 2080.
if N is a non-negative integer that can be written as a²+520b², then p × N can be written as a²+520b²
if N is a non-negative integer that cannot be written as a²+520b², then p × N cannot be written as a²+520b²

Class A primes: All primes congruent to [5, 21, 109, 141, 149, 189, 229, 301, 349, 421, 461, 501, 509, 541, 629, 661, 669, 709, 749, 821, 869, 941, 981, 1021, 1029, 1061, 1149, 1181, 1189, 1229, 1269, 1341, 1389, 1461, 1501, 1541, 1549, 1581, 1669, 1701, 1709, 1749, 1789, 1861, 1909, 1981, 2021, 2061, 2069] mod 2080
Class B primes: All primes congruent to [13, 53, 77, 133, 157, 173, 237, 277, 373, 413, 477, 493, 517, 573, 597, 653, 677, 693, 757, 797, 893, 933, 997, 1013, 1037, 1093, 1117, 1173, 1197, 1213, 1277, 1317, 1413, 1453, 1517, 1533, 1557, 1613, 1637, 1693, 1717, 1733, 1797, 1837, 1933, 1973, 2037, 2053, 2077] mod 2080
Class C primes: All primes congruent to [23, 87, 103, 127, 183, 207, 263, 287, 303, 367, 407, 503, 543, 607, 623, 647, 703, 727, 783, 807, 823, 887, 927, 1023, 1063, 1127, 1143, 1167, 1223, 1247, 1303, 1327, 1343, 1407, 1447, 1543, 1583, 1647, 1663, 1687, 1743, 1767, 1823, 1847, 1863, 1927, 1967, 2063] mod 2080
Class D primes: All primes congruent to [31, 71, 111, 119, 151, 239, 271, 279, 319, 359, 431, 479, 551, 591, 631, 639, 671, 759, 791, 799, 839, 879, 951, 999, 1071, 1111, 1151, 1159, 1191, 1279, 1311, 1319, 1359, 1399, 1471, 1519, 1591, 1631, 1671, 1679, 1711, 1799, 1831, 1839, 1879, 1919, 1991, 2039] mod 2080
Class E primes: All primes congruent to [67, 83, 123, 163, 187, 203, 227, 267, 307, 323, 427, 483, 587, 603, 643, 683, 707, 723, 747, 787, 827, 843, 947, 1003, 1107, 1123, 1163, 1203, 1227, 1243, 1267, 1307, 1347, 1363, 1467, 1523, 1627, 1643, 1683, 1723, 1747, 1763, 1787, 1827, 1867, 1883, 1987, 2043] mod 2080
Class F primes: All primes congruent to [33, 57, 73, 97, 137, 177, 193, 297, 353, 457, 473, 513, 553, 577, 593, 617, 657, 697, 713, 817, 873, 977, 993, 1033, 1073, 1097, 1113, 1137, 1177, 1217, 1233, 1337, 1393, 1497, 1513, 1553, 1593, 1617, 1633, 1657, 1697, 1737, 1753, 1857, 1913, 2017, 2033, 2073] mod 2080
Class G primes: All primes congruent to [51, 131, 139, 179, 211, 251, 259, 339, 419, 451, 459, 491, 571, 651, 659, 699, 731, 771, 779, 859, 939, 971, 979, 1011, 1091, 1171, 1179, 1219, 1251, 1291, 1299, 1379, 1459, 1491, 1499, 1531, 1611, 1691, 1699, 1739, 1771, 1811, 1819, 1899, 1979, 2011, 2019, 2051] mod 2080

N can be written as a²+520b² if and only if
- N is not congruent to 2 (mod 4).
- N has no prime factors congruent to [3, 7, 11, 17, 19, 27, 29, 37, 41, 43, 47, 59, 61, 63, 69, 79, 89, 93, 99, 101, 107, 113, 147, 153, 159, 161, 167, 171, 181, 191, 197, 199, 201, 213, 217, 219, 223, 231, 233, 241, 243, 249, 253, 257, 261, 269, 281, 283, 291, 293, 309, 311, 313, 317, 327, 331, 333, 337, 341, 343, 347, 357, 363, 369, 371, 379, 381, 383, 387, 389, 391, 393, 397, 399, 401, 409, 411, 417, 423, 433, 437, 439, 443, 447, 449, 453, 463, 467, 469, 471, 487, 489, 497, 499, 511, 519, 523, 527, 531, 537, 539, 547, 549, 557, 561, 563, 567, 579, 581, 583, 589, 599, 609, 613, 619, 621, 627, 633, 667, 673, 679, 681, 687, 691, 701, 711, 717, 719, 721, 733, 737, 739, 743, 751, 753, 761, 763, 769, 773, 777, 781, 789, 801, 803, 811, 813, 829, 831, 833, 837, 847, 851, 853, 857, 861, 863, 867, 877, 883, 889, 891, 899, 901, 903, 907, 909, 911, 913, 917, 919, 921, 929, 931, 937, 943, 953, 957, 959, 963, 967, 969, 973, 983, 987, 989, 991, 1007, 1009, 1017, 1019, 1031, 1039, 1043, 1047, 1051, 1057, 1059, 1067, 1069, 1077, 1081, 1083, 1087, 1099, 1101, 1103, 1109, 1119, 1129, 1133, 1139, 1141, 1147, 1153, 1187, 1193, 1199, 1201, 1207, 1211, 1221, 1231, 1237, 1239, 1241, 1253, 1257, 1259, 1263, 1271, 1273, 1281, 1283, 1289, 1293, 1297, 1301, 1309, 1321, 1323, 1331, 1333, 1349, 1351, 1353, 1357, 1367, 1371, 1373, 1377, 1381, 1383, 1387, 1397, 1403, 1409, 1411, 1419, 1421, 1423, 1427, 1429, 1431, 1433, 1437, 1439, 1441, 1449, 1451, 1457, 1463, 1473, 1477, 1479, 1483, 1487, 1489, 1493, 1503, 1507, 1509, 1511, 1527, 1529, 1537, 1539, 1551, 1559, 1563, 1567, 1571, 1577, 1579, 1587, 1589, 1597, 1601, 1603, 1607, 1619, 1621, 1623, 1629, 1639, 1649, 1653, 1659, 1661, 1667, 1673, 1707, 1713, 1719, 1721, 1727, 1731, 1741, 1751, 1757, 1759, 1761, 1773, 1777, 1779, 1783, 1791, 1793, 1801, 1803, 1809, 1813, 1817, 1821, 1829, 1841, 1843, 1851, 1853, 1869, 1871, 1873, 1877, 1887, 1891, 1893, 1897, 1901, 1903, 1907, 1917, 1923, 1929, 1931, 1939, 1941, 1943, 1947, 1949, 1951, 1953, 1957, 1959, 1961, 1969, 1971, 1977, 1983, 1993, 1997, 1999, 2003, 2007, 2009, 2013, 2023, 2027, 2029, 2031, 2047, 2049, 2057, 2059, 2071, 2079] mod 2080 to an odd power.
- If N is odd, then the sum of exponents of the prime factors of N of the classes A,B,C,D,E,F,G are all simultaneously odd, or the sum of exponents of the prime factors of N of the classes A,B,C,D,E,F,G are all simultaneously even, or the sum of exponents of the prime factors of N of these following classes {A,B,C,D or A,C,E or B,D,E or A,B,F or C,D,F or B,C,E,F or A,D,E,F or B,C,G or A,D,G or A,B,E,G or C,D,E,G or A,C,F,G or B,D,F,G or E,F,G} are all simultaneously odd, with the sum of exponents of the prime factors of N of the rest of the classes are all simultaneously even.
- If the highest power of 2 dividing N is an even number that is two or more, then the sum of exponents of the prime factors of N of the classes A,B,C,D,E,F,G are all simultaneously odd, or the sum of exponents of the prime factors of N of the classes A,B,C,D,E,F,G are all simultaneously even, or the sum of exponents of the prime factors of N of these following classes {B,C or A,D or A,B,C,D or A,B,E or A,C,E or B,D,E or C,D,E or A,B,F or A,C,F or B,D,F or C,D,F or E,F or B,C,E,F or A,D,E,F or A,B,C,D,E,F or G or B,C,G or A,D,G or A,B,C,D,G or A,B,E,G or A,C,E,G or B,D,E,G or C,D,E,G or A,B,F,G or A,C,F,G or B,D,F,G or C,D,F,G or E,F,G or B,C,E,F,G or A,D,E,F,G} are all simultaneously odd, with the sum of exponents of the prime factors of N of the rest of the classes are all simultaneously even.
- If the highest power of 2 dividing N is an odd number that is three or more, then the sum of exponents of the prime factors of N of these following classes {A,B or A,C or B,D or C,D or E or B,C,E or A,D,E or A,B,C,D,E or F or B,C,F or A,D,F or A,B,C,D,F or A,B,E,F or A,C,E,F or B,D,E,F or C,D,E,F or A,B,G or A,C,G or B,D,G or C,D,G or E,G or B,C,E,G or A,D,E,G or A,B,C,D,E,G or F,G or B,C,F,G or A,D,F,G or A,B,C,D,F,G or A,B,E,F,G or A,C,E,F,G or B,D,E,F,G or C,D,E,F,G} are all simultaneously odd, with the sum of exponents of the prime factors of N of the rest of the classes are all simultaneously even.

[Raman]

k = 760

Primes p of the form a²+760b²: All primes congruent to [1, 9, 49, 81, 121, 161, 169, 201, 289, 321, 329, 441, 481, 529, 609, 681, 689, 729, 761, 769, 809, 841, 881, 921, 929, 961, 1049, 1081, 1089, 1201, 1241, 1289, 1369, 1441, 1449, 1489, 1521, 1529, 1569, 1601, 1641, 1681, 1689, 1721, 1809, 1841, 1849, 1961, 2001, 2049, 2129, 2201, 2209, 2249, 2281, 2289, 2329, 2361, 2401, 2441, 2449, 2481, 2569, 2601, 2609, 2721, 2761, 2809, 2889, 2961, 2969, 3009] mod 3040.
if N is a non-negative integer that can be written as a²+760b², then p × N can be written as a²+760b²
if N is a non-negative integer that cannot be written as a²+760b², then p × N cannot be written as a²+760b²

Class A primes: All primes congruent to [5, 77, 93, 157, 197, 213, 237, 253, 277, 397, 453, 517, 533, 557, 613, 653, 693, 733, 757, 837, 853, 917, 957, 973, 997, 1013, 1037, 1157, 1213, 1277, 1293, 1317, 1373, 1413, 1453, 1493, 1517, 1597, 1613, 1677, 1717, 1733, 1757, 1773, 1797, 1917, 1973, 2037, 2053, 2077, 2133, 2173, 2213, 2253, 2277, 2357, 2373, 2437, 2477, 2493, 2517, 2533, 2557, 2677, 2733, 2797, 2813, 2837, 2893, 2933, 2973, 3013, 3037] mod 3040
Class B primes: All primes congruent to [19, 51, 59, 91, 179, 211, 219, 259, 299, 331, 371, 379, 411, 451, 459, 531, 611, 659, 699, 811, 819, 851, 939, 971, 979, 1019, 1059, 1091, 1131, 1139, 1171, 1211, 1219, 1291, 1371, 1419, 1459, 1571, 1579, 1611, 1699, 1731, 1739, 1779, 1819, 1851, 1891, 1899, 1931, 1971, 1979, 2051, 2131, 2179, 2219, 2331, 2339, 2371, 2459, 2491, 2499, 2539, 2579, 2611, 2651, 2659, 2691, 2731, 2739, 2811, 2891, 2939, 2979] mod 3040
Class C primes: All primes congruent to [21, 29, 69, 109, 141, 181, 189, 221, 261, 269, 341, 421, 469, 509, 621, 629, 661, 749, 781, 789, 829, 869, 901, 941, 949, 981, 1021, 1029, 1101, 1181, 1229, 1269, 1381, 1389, 1421, 1509, 1541, 1549, 1589, 1629, 1661, 1701, 1709, 1741, 1781, 1789, 1861, 1941, 1989, 2029, 2141, 2149, 2181, 2269, 2301, 2309, 2349, 2389, 2421, 2461, 2469, 2501, 2541, 2549, 2621, 2701, 2749, 2789, 2901, 2909, 2941, 3029] mod 3040
Class D primes: All primes congruent to [43, 83, 123, 163, 187, 267, 283, 347, 387, 403, 427, 443, 467, 587, 643, 707, 723, 747, 803, 843, 883, 923, 947, 1027, 1043, 1107, 1147, 1163, 1187, 1203, 1227, 1347, 1403, 1467, 1483, 1507, 1563, 1603, 1643, 1683, 1707, 1787, 1803, 1867, 1907, 1923, 1947, 1963, 1987, 2107, 2163, 2227, 2243, 2267, 2323, 2363, 2403, 2443, 2467, 2547, 2563, 2627, 2667, 2683, 2707, 2723, 2747, 2867, 2923, 2987, 3003, 3027] mod 3040
Class E primes: All primes congruent to [33, 97, 113, 193, 217, 257, 297, 337, 393, 417, 433, 497, 553, 673, 697, 713, 737, 753, 793, 857, 873, 953, 977, 1017, 1057, 1097, 1153, 1177, 1193, 1257, 1313, 1433, 1457, 1473, 1497, 1513, 1553, 1617, 1633, 1713, 1737, 1777, 1817, 1857, 1913, 1937, 1953, 2017, 2073, 2193, 2217, 2233, 2257, 2273, 2313, 2377, 2393, 2473, 2497, 2537, 2577, 2617, 2673, 2697, 2713, 2777, 2833, 2953, 2977, 2993, 3017, 3033] mod 3040
Class F primes: All primes congruent to [103, 127, 143, 167, 183, 223, 287, 303, 383, 407, 447, 487, 527, 583, 607, 623, 687, 743, 863, 887, 903, 927, 943, 983, 1047, 1063, 1143, 1167, 1207, 1247, 1287, 1343, 1367, 1383, 1447, 1503, 1623, 1647, 1663, 1687, 1703, 1743, 1807, 1823, 1903, 1927, 1967, 2007, 2047, 2103, 2127, 2143, 2207, 2263, 2383, 2407, 2423, 2447, 2463, 2503, 2567, 2583, 2663, 2687, 2727, 2767, 2807, 2863, 2887, 2903, 2967, 3023] mod 3040
Class G primes: All primes congruent to [39, 111, 119, 159, 191, 199, 239, 271, 311, 351, 359, 391, 479, 511, 519, 631, 671, 719, 799, 871, 879, 919, 951, 959, 999, 1031, 1071, 1111, 1119, 1151, 1239, 1271, 1279, 1391, 1431, 1479, 1559, 1631, 1639, 1679, 1711, 1719, 1759, 1791, 1831, 1871, 1879, 1911, 1999, 2031, 2039, 2151, 2191, 2239, 2319, 2391, 2399, 2439, 2471, 2479, 2519, 2551, 2591, 2631, 2639, 2671, 2759, 2791, 2799, 2911, 2951, 2999] mod 3040

N can be written as a²+760b² if and only if
- N is not congruent to 2 (mod 4).
- N has no prime factors congruent to [3, 7, 11, 13, 17, 23, 27, 31, 37, 41, 47, 53, 61, 63, 67, 71, 73, 79, 87, 89, 99, 101, 107, 117, 129, 131, 137, 139, 147, 149, 151, 153, 173, 177, 203, 207, 227, 229, 231, 233, 241, 243, 249, 251, 263, 273, 279, 281, 291, 293, 301, 307, 309, 313, 317, 319, 327, 333, 339, 343, 349, 353, 357, 363, 367, 369, 373, 377, 381, 389, 401, 409, 413, 419, 423, 429, 431, 439, 449, 457, 461, 463, 471, 473, 477, 483, 489, 491, 493, 499, 501, 503, 507, 521, 523, 537, 539, 541, 543, 547, 549, 559, 561, 563, 567, 569, 571, 573, 577, 579, 581, 591, 593, 597, 599, 601, 603, 617, 619, 633, 637, 639, 641, 647, 649, 651, 657, 663, 667, 669, 677, 679, 683, 691, 701, 709, 711, 717, 721, 727, 731, 739, 751, 759, 763, 767, 771, 773, 777, 783, 787, 791, 797, 801, 807, 813, 821, 823, 827, 831, 833, 839, 847, 849, 859, 861, 867, 877, 889, 891, 897, 899, 907, 909, 911, 913, 933, 937, 963, 967, 987, 989, 991, 993, 1001, 1003, 1009, 1011, 1023, 1033, 1039, 1041, 1051, 1053, 1061, 1067, 1069, 1073, 1077, 1079, 1087, 1093, 1099, 1103, 1109, 1113, 1117, 1123, 1127, 1129, 1133, 1137, 1141, 1149, 1161, 1169, 1173, 1179, 1183, 1189, 1191, 1199, 1209, 1217, 1221, 1223, 1231, 1233, 1237, 1243, 1249, 1251, 1253, 1259, 1261, 1263, 1267, 1281, 1283, 1297, 1299, 1301, 1303, 1307, 1309, 1319, 1321, 1323, 1327, 1329, 1331, 1333, 1337, 1339, 1341, 1351, 1353, 1357, 1359, 1361, 1363, 1377, 1379, 1393, 1397, 1399, 1401, 1407, 1409, 1411, 1417, 1423, 1427, 1429, 1437, 1439, 1443, 1451, 1461, 1469, 1471, 1477, 1481, 1487, 1491, 1499, 1511, 1519, 1523, 1527, 1531, 1533, 1537, 1543, 1547, 1551, 1557, 1561, 1567, 1573, 1581, 1583, 1587, 1591, 1593, 1599, 1607, 1609, 1619, 1621, 1627, 1637, 1649, 1651, 1657, 1659, 1667, 1669, 1671, 1673, 1693, 1697, 1723, 1727, 1747, 1749, 1751, 1753, 1761, 1763, 1769, 1771, 1783, 1793, 1799, 1801, 1811, 1813, 1821, 1827, 1829, 1833, 1837, 1839, 1847, 1853, 1859, 1863, 1869, 1873, 1877, 1883, 1887, 1889, 1893, 1897, 1901, 1909, 1921, 1929, 1933, 1939, 1943, 1949, 1951, 1959, 1969, 1977, 1981, 1983, 1991, 1993, 1997, 2003, 2009, 2011, 2013, 2019, 2021, 2023, 2027, 2041, 2043, 2057, 2059, 2061, 2063, 2067, 2069, 2079, 2081, 2083, 2087, 2089, 2091, 2093, 2097, 2099, 2101, 2111, 2113, 2117, 2119, 2121, 2123, 2137, 2139, 2153, 2157, 2159, 2161, 2167, 2169, 2171, 2177, 2183, 2187, 2189, 2197, 2199, 2203, 2211, 2221, 2229, 2231, 2237, 2241, 2247, 2251, 2259, 2271, 2279, 2283, 2287, 2291, 2293, 2297, 2303, 2307, 2311, 2317, 2321, 2327, 2333, 2341, 2343, 2347, 2351, 2353, 2359, 2367, 2369, 2379, 2381, 2387, 2397, 2409, 2411, 2417, 2419, 2427, 2429, 2431, 2433, 2453, 2457, 2483, 2487, 2507, 2509, 2511, 2513, 2521, 2523, 2529, 2531, 2543, 2553, 2559, 2561, 2571, 2573, 2581, 2587, 2589, 2593, 2597, 2599, 2607, 2613, 2619, 2623, 2629, 2633, 2637, 2643, 2647, 2649, 2653, 2657, 2661, 2669, 2681, 2689, 2693, 2699, 2703, 2709, 2711, 2719, 2729, 2737, 2741, 2743, 2751, 2753, 2757, 2763, 2769, 2771, 2773, 2779, 2781, 2783, 2787, 2801, 2803, 2817, 2819, 2821, 2823, 2827, 2829, 2839, 2841, 2843, 2847, 2849, 2851, 2853, 2857, 2859, 2861, 2871, 2873, 2877, 2879, 2881, 2883, 2897, 2899, 2913, 2917, 2919, 2921, 2927, 2929, 2931, 2937, 2943, 2947, 2949, 2957, 2959, 2963, 2971, 2981, 2989, 2991, 2997, 3001, 3007, 3011, 3019, 3031, 3039] mod 3040 to an odd power.
- If N is odd, then the sum of exponents of the prime factors of N of the classes A,B,C,D,E,F,G are all simultaneously odd, or the sum of exponents of the prime factors of N of the classes A,B,C,D,E,F,G are all simultaneously even, or the sum of exponents of the prime factors of N of these following classes {A,B,C,D or A,C,E or B,D,E or A,B,F or C,D,F or B,C,E,F or A,D,E,F or B,C,G or A,D,G or A,B,E,G or C,D,E,G or A,C,F,G or B,D,F,G or E,F,G} are all simultaneously odd, with the sum of exponents of the prime factors of N of the rest of the classes are all simultaneously even.
- If the highest power of 2 dividing N is an even number that is two or more, then the sum of exponents of the prime factors of N of the classes A,B,C,D,E,F,G are all simultaneously odd, or the sum of exponents of the prime factors of N of the classes A,B,C,D,E,F,G are all simultaneously even, or the sum of exponents of the prime factors of N of these following classes {B,C or A,D or A,B,C,D or A,B,E or A,C,E or B,D,E or C,D,E or A,B,F or A,C,F or B,D,F or C,D,F or E,F or B,C,E,F or A,D,E,F or A,B,C,D,E,F or G or B,C,G or A,D,G or A,B,C,D,G or A,B,E,G or A,C,E,G or B,D,E,G or C,D,E,G or A,B,F,G or A,C,F,G or B,D,F,G or C,D,F,G or E,F,G or B,C,E,F,G or A,D,E,F,G} are all simultaneously odd, with the sum of exponents of the prime factors of N of the rest of the classes are all simultaneously even.
- If the highest power of 2 dividing N is an odd number that is three or more, then the sum of exponents of the prime factors of N of these following classes {A,B or A,C or B,D or C,D or E or B,C,E or A,D,E or A,B,C,D,E or F or B,C,F or A,D,F or A,B,C,D,F or A,B,E,F or A,C,E,F or B,D,E,F or C,D,E,F or A,B,G or A,C,G or B,D,G or C,D,G or E,G or B,C,E,G or A,D,E,G or A,B,C,D,E,G or F,G or B,C,F,G or A,D,F,G or A,B,C,D,F,G or A,B,E,F,G or A,C,E,F,G or B,D,E,F,G or C,D,E,F,G} are all simultaneously odd, with the sum of exponents of the prime factors of N of the rest of the classes are all simultaneously even.

[Xyzzy]