307 and pg primes

enzocreti · 2019-10-11, 07:55

pg(215), pg(69660), pg(92020) and pg(541456) are probable primes

215, 69660, 92020 and 541456 are multiples of 43

1456=307*8-1000

92020-(307*8-1000)+1 is a multiple of 307
5414560-1 is a multiple of 307
2150-1 is a multiple of 307

so 92020 is congruent to (307*8-1001) mod 307

215 is congruent to 170*1455 mod 307
69660 is congruent to 127*1455 mod 307
92020 is congruent to 1*1455 mod 307
541456 is congruent to 170*1455 mod 307

170,127 are congruent to 41 mod 43

so 215 69660 92020 541456 are of the form 41*43+r
where r is in the set (215,344,903)
and 215,69660,92020,541456 are congruent to 1455*s mod 307
where s=1 or s is of the form 43m+41

so trying to simplifying:

215, 69660, 92020 and 541456 are either congruent to (41*43-308) mod 307 or congruent to (43m+41)*(41*43-308) mod 307 with m some integer

...expanding (43m+41)*(41*43-308) you get

m(43^2*41-43*308)+41^2*43-41*308

122221=41*43^2-43*308+43*41^2-41*308+1

308=(41*43+1009)/9 where 1009=10^3+3^2 is prime

so the minimum value j such that

215-j*1455
69660-j*1455
92020-j*1455
541456-j*1455
are congruent to 0 mod 307 is either j=1 or j an integer of the form 43s+41

215 and 541456 are both congruent to 215 mod 307
92020 is congruent to 227 mod 307
but also 41*43-1=1762 is congruent to 227 mod 307
69660 is congruent to 278 mod 307
but also 17163=41*43*9+6^4 is congruent to 278 mod 307

69660 is of the form 1763*s+903

903=int(((1763*9+6^4)/19) where int is the integer part and 17163=1763*9+6^4

69660=19*(1763+10^3)+17163=19*(307*9)+17163

(1763*3+2*216)/2-1763*3/2-1=215

Is there any reason for this:

69660, 92020, 541456 are congruent to 307*(4+13*s) mod 1456 where s is a nonnegative integer

infact 69660 is congruent to 307*4 mod 1456
92020 is congruent to 307*(4+13*72) mod 1456
541456 is congruent to 307*(4+13*108) mod 1456

92020 mod 307 is 227=307*4-1001=1763 mod 307 -1=228-1

69660 mod 1456 is 307*4=227+1001
541456 mod 1456 is 1280=69660 mod 307+1002.
((1763 mod 307)+1000)*2-1000=1456
69660 mod 1456=(1763 mod 307)+1000

69660 mod 1456 =1228=1456-(1763 mod 307)

so 69660 is congruent to (1763 mod 307)*2+10^3 mod 1456

215=(92020-1000-228)/13/8-(69660-1000-228)/13/8
where 1228 is 69660 mod 1456

215 (odd) is congruent to (1763 mod 307)=228 mod 13
69660 (even) +1 is congruent to (1763 mod 307) mod 13
92020 (even) +1 is congruent to (1763 mod 307) mod 13
541456 (even)+1 is congruent to (1763 mod 307) mod 13

(1763 mod 307)-1=227 is a prime

so 215 is 228 mod 13
69660 is 227 mod 13
92020 is 227 mod 13
541456 is 227 mod 13

Incidentally 92020 is also 227 mod 307

because 13=1456/112
the above can be rewritten as

215 is 228 mod ((1763-307)/112)
69660 is 227 mod ((1763-307)/112)
92020 is 227 mod ((1763-307)/112)
541456 is 227 mod ((1763-307)/112)

113*2+1=227 is a Sophie germain prime

because ((1763 mod 307)+1000)*2-1000=1456

215 is 228 mod (((1763 mod 307)+1000)*2-1000)/112...

another way is
215 is congruent to (42^2-3*2^9) mod (3*2^2+1)

69660 is congruent to (1763-3*2^9) mod (3*2^2+1)
92020 is congruent to (1763-3*2^9) mod (3*2^2+1)
541456 is congruent to (1763-3*2^9) mod (3*2^2+1)

where 9 is the order of 3*2^q mod 307 because 9 is the least value of q such that 3*2^q is congruent to 1 mod 307

1763 mod 307=228=(307*9-1000) mod 307

1456 is 10^3+ the sum of two twin primes 227 and 229...1456=10^3+227+229

228 is the residue mod 10^3 of 307*2^2
1000 is the residue mod 1763 of 307*3^2

the numbers either have the form 1763*(13s)+r as in the case of 69660=1763*39+903 and 92020=1763*52+344 or the form 1763*(307s)+r as in the case of 215 (s=0,r=215) and 541456

92020 has also the representation 92020=307*13*23+227

and 1763 and 307 have the same residue 2^3 mod 13

so 92020=(307-8)*13+227

so the numbers can be written if even:

N=(13x+1762-307y)

for example

69660=(429*13+1762-307*(-203))

69660=13x-307y+1762 solution x=429 y=-203
92020=13x-307y+1762 solution x=307 y=281
541456=13x-307y+1762 solution x=495 y=1737

take the absolute value of the y's: 203,281,1737 are congruent to 8 mod 13

so the numbers have the form 13x-307*(13y+8)*z+1762
with x,y,z some integers
this is the equation of a hyperbolic paraboloid

215 = 13 x - 3991 y z - 2456 y + 1762

Batalov · 2019-10-11, 08:11

You are posting essentially one and the same post about a hundred times now. Again and again. In a new thread each time, no less. It is truly annoying.

Once every three months or so, we will try to gently ask you to stop your obsessive compulsive posting behavior. Maybe one day you will listen? It will do you a lot of good. The rest of the time we will not even look. Do what you want.

2019-10-11, 07:55	#1
enzocreti Mar 2018 530₁₀ Posts	307 and pg primes pg(215), pg(69660), pg(92020) and pg(541456) are probable primes 215, 69660, 92020 and 541456 are multiples of 43 1456=3078-1000 92020-(3078-1000)+1 is a multiple of 307 5414560-1 is a multiple of 307 2150-1 is a multiple of 307 so 92020 is congruent to (3078-1001) mod 307 215 is congruent to 1701455 mod 307 69660 is congruent to 1271455 mod 307 92020 is congruent to 11455 mod 307 541456 is congruent to 1701455 mod 307 170,127 are congruent to 41 mod 43 so 215 69660 92020 541456 are of the form 4143+r where r is in the set (215,344,903) and 215,69660,92020,541456 are congruent to 1455s mod 307 where s=1 or s is of the form 43m+41 so trying to simplifying: 215, 69660, 92020 and 541456 are either congruent to (4143-308) mod 307 or congruent to (43m+41)(4143-308) mod 307 with m some integer ...expanding (43m+41)(4143-308) you get m(43^241-43308)+41^243-41308 122221=4143^2-43308+4341^2-41308+1 308=(4143+1009)/9 where 1009=10^3+3^2 is prime so the minimum value j such that 215-j1455 69660-j1455 92020-j1455 541456-j1455 are congruent to 0 mod 307 is either j=1 or j an integer of the form 43s+41 215 and 541456 are both congruent to 215 mod 307 92020 is congruent to 227 mod 307 but also 4143-1=1762 is congruent to 227 mod 307 69660 is congruent to 278 mod 307 but also 17163=41439+6^4 is congruent to 278 mod 307 69660 is of the form 1763s+903 903=int(((17639+6^4)/19) where int is the integer part and 17163=17639+6^4 69660=19(1763+10^3)+17163=19(3079)+17163 (17633+2216)/2-17633/2-1=215 Is there any reason for this: 69660, 92020, 541456 are congruent to 307(4+13s) mod 1456 where s is a nonnegative integer infact 69660 is congruent to 3074 mod 1456 92020 is congruent to 307(4+1372) mod 1456 541456 is congruent to 307(4+13108) mod 1456 92020 mod 307 is 227=3074-1001=1763 mod 307 -1=228-1 69660 mod 1456 is 3074=227+1001 541456 mod 1456 is 1280=69660 mod 307+1002. ((1763 mod 307)+1000)2-1000=1456 69660 mod 1456=(1763 mod 307)+1000 69660 mod 1456 =1228=1456-(1763 mod 307) so 69660 is congruent to (1763 mod 307)2+10^3 mod 1456 215=(92020-1000-228)/13/8-(69660-1000-228)/13/8 where 1228 is 69660 mod 1456 215 (odd) is congruent to (1763 mod 307)=228 mod 13 69660 (even) +1 is congruent to (1763 mod 307) mod 13 92020 (even) +1 is congruent to (1763 mod 307) mod 13 541456 (even)+1 is congruent to (1763 mod 307) mod 13 (1763 mod 307)-1=227 is a prime so 215 is 228 mod 13 69660 is 227 mod 13 92020 is 227 mod 13 541456 is 227 mod 13 Incidentally 92020 is also 227 mod 307 because 13=1456/112 the above can be rewritten as 215 is 228 mod ((1763-307)/112) 69660 is 227 mod ((1763-307)/112) 92020 is 227 mod ((1763-307)/112) 541456 is 227 mod ((1763-307)/112) 1132+1=227 is a Sophie germain prime because ((1763 mod 307)+1000)2-1000=1456 215 is 228 mod (((1763 mod 307)+1000)2-1000)/112... another way is 215 is congruent to (42^2-32^9) mod (32^2+1) 69660 is congruent to (1763-32^9) mod (32^2+1) 92020 is congruent to (1763-32^9) mod (32^2+1) 541456 is congruent to (1763-32^9) mod (32^2+1) where 9 is the order of 32^q mod 307 because 9 is the least value of q such that 32^q is congruent to 1 mod 307 1763 mod 307=228=(3079-1000) mod 307 1456 is 10^3+ the sum of two twin primes 227 and 229...1456=10^3+227+229 228 is the residue mod 10^3 of 3072^2 1000 is the residue mod 1763 of 3073^2 the numbers either have the form 1763(13s)+r as in the case of 69660=176339+903 and 92020=176352+344 or the form 1763(307s)+r as in the case of 215 (s=0,r=215) and 541456 92020 has also the representation 92020=3071323+227 and 1763 and 307 have the same residue 2^3 mod 13 so 92020=(307-8)13+227 so the numbers can be written if even: N=(13x+1762-307y) for example 69660=(42913+1762-307(-203)) 69660=13x-307y+1762 solution x=429 y=-203 92020=13x-307y+1762 solution x=307 y=281 541456=13x-307y+1762 solution x=495 y=1737 take the absolute value of the y's: 203,281,1737 are congruent to 8 mod 13 so the numbers have the form 13x-307(13y+8)z+1762 with x,y,z some integers this is the equation of a hyperbolic paraboloid 215 = 13 x - 3991 y z - 2456 y + 1762 Last fiddled with by enzocreti on 2019-10-20 at 12:27*

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2019-10-11, 08:11	#2
Batalov "Serge" Mar 2008 Phi(4,2^7658614+1)/2 3⁶·13 Posts	You are posting essentially one and the same post about a hundred times now. Again and again. In a new thread each time, no less. It is truly annoying. Once every three months or so, we will try to gently ask you to stop your obsessive compulsive posting behavior. Maybe one day you will listen? It will do you a lot of good. The rest of the time we will not even look. Do what you want.