## GFN3 conjecture

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• Conjecture: If N = 3*8^n+1 is prime, then 3^((N-1)/6) = 1 mod N. Comment: We know of 35 primes of this form, namely those with n = 2,4,6,10,12,22,63,67,92,136,
Message 1 of 5 , Feb 2, 2014
Conjecture: If N = 3*8^n+1 is prime, then 3^((N-1)/6) = 1 mod N.

Comment: We know of 35 primes of this form, namely those with
n = 2,4,6,10,12,22,63,67,92,136,
146,178,736,1056,1063,1304,11450,14098,14895,16050,
18264,18394,19991,26730,71107,101031,127483,236656,267326,305591,
610832,763870,1694102,2344547,3609782
and each has 3^((N-1)/6) = 1 mod N, which is equivalent
to saying that N is a divisor of GF(m,3) = 3^(2^m) + 1
for some m < 3*n - 1.

David
• For prime p=1 (mod 12) we know 3 to be a quadratic residue mod p. For prime p=3k^3+1 number 3 is a cubic residue mod p, because 3=(-1/k)^3 (mod p). Since k is
Message 2 of 5 , Feb 2, 2014
For prime p=1 (mod 12) we know 3 to be a quadratic residue mod p.

For prime p=3k^3+1 number 3 is a cubic residue mod p, because 3=(-1/k)^3 (mod p).
Since k is even, we have also p=1 (mod 12) and 3 is a quadratic residue mod p.
Hence 3 is a 6th power residue, i.e 3^((p-1)/6) = 1 (mod p).

Jarek

2014-02-02 :

Conjecture: If N = 3*8^n+1 is prime, then 3^((N-1)/6) = 1 mod N.

Comment: We know of 35 primes of this form, namely those with
n = 2,4,6,10,12,22,63,67,92,136,
146,178,736,1056,1063,1304,11450,14098,14895,16050,
18264,18394,19991,26730,71107,101031,127483,236656,267326,305591,
610832,763870,1694102,2344547,3609782
and each has 3^((N-1)/6) = 1 mod N, which is equivalent
to saying that N is a divisor of GF(m,3) = 3^(2^m) + 1
for some m < 3*n - 1.

David

• ... That s great, Jarek! Just want I wanted and was not able to prove by myself. Dziękuję bardzo! David 2014-02-02
Message 3 of 5 , Feb 2, 2014
Jaroslaw Wroblewski wrote:

> For prime p=1 (mod 12) we know 3 to be a quadratic residue mod p.
> For prime p=3k^3+1 number 3 is a cubic residue mod p, because
> 3=(-1/k)^3 (mod p).
> Since k is even, we have also p=1 (mod 12)
> and 3 is a quadratic residue mod p.
> Hence 3 is a 6th power residue, i.e 3^((p-1)/6) = 1 (mod p).

That's great, Jarek!

Just want I wanted and was not able to prove by myself.

Dziękuję bardzo!

David

2014-02-02 :

Conjecture: If N = 3*8^n+1 is prime, then 3^((N-1)/6) = 1 mod N.

Comment: We know of 35 primes of this form, namely those with
n = 2,4,6,10,12,22,63,67,92,136,
146,178,736,1056,1063,1304,11450,14098,14895,16050,
18264,18394,19991,26730,71107,101031,127483,236656,267326,305591,
610832,763870,1694102,2344547,3609782
and each has 3^((N-1)/6) = 1 mod N, which is equivalent
to saying that N is a divisor of GF(m,3) = 3^(2^m) + 1
for some m < 3*n - 1.

David

• Hi every body . My theory to finding primes will be the best theory in number theory . I can find any prime number by TMSM theory . Thamer M . Masarweh ‏من
Message 4 of 5 , Feb 3, 2014
Hi every body . My theory to finding primes will be the best theory in number theory .

I can find any prime number by TMSM theory .

Thamer M . Masarweh

‏من جهاز الـ iPad الخاص بي

في ٠٣‏/٠٢‏/٢٠١٤، الساعة ٥:٣٦ ص، كتب Jaroslaw Wroblewski <jaroslaw.wroblewski@...>:

For prime p=1 (mod 12) we know 3 to be a quadratic residue mod p.

For prime p=3k^3+1 number 3 is a cubic residue mod p, because 3=(-1/k)^3 (mod p).
Since k is even, we have also p=1 (mod 12) and 3 is a quadratic residue mod p.
Hence 3 is a 6th power residue, i.e 3^((p-1)/6) = 1 (mod p).

Jarek

2014-02-02 :

Conjecture: If N = 3*8^n+1 is prime, then 3^((N-1)/6) = 1 mod N.

Comment: We know of 35 primes of this form, namely those with
n = 2,4,6,10,12,22,63,67,92,136,
146,178,736,1056,1063,1304,11450,14098,14895,16050,
18264,18394,19991,26730,71107,101031,127483,236656,267326,305591,
610832,763870,1694102,2344547,3609782
and each has 3^((N-1)/6) = 1 mod N, which is equivalent
to saying that N is a divisor of GF(m,3) = 3^(2^m) + 1
for some m < 3*n - 1.

David

• ... It is now some time since he and his brother, Ragheb, had their profiles in the Jordan Times. Until Thamer has something of substance to communicate,
Message 5 of 5 , Feb 3, 2014

Thamer Masarweh spammed:

> I can find any prime number by TMSM theory

It is now some time since he and his brother, Ragheb,
had their profiles in the Jordan Times.

Until Thamer has something of substance to communicate,
regarding primality, he might refrain from posting to this list?

David (with condolences to Jarek, whose proof was spammed)
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