A269454 Safe primes that are not congruent to -1 mod 8.

5, 11, 59, 83, 107, 179, 227, 347, 467, 563, 587, 1019, 1187, 1283, 1307, 1523, 1619, 1907, 2027, 2099, 2459, 2579, 2819, 2963, 3203, 3467, 3779, 3803, 3947, 4139, 4259, 4283, 4547, 4787, 5099, 5387, 5483, 5507, 5939, 6659, 6779, 6827, 6899, 7187, 7523

Offset: 1

Author

Marina Ibrishimova, Feb 27 2016

Keywords

• nonn

Comments

For safe primes see A005385.

Conjecture: If p and q are two distinct safe primes not congruent to -1 mod 8 then the order of 2 mod p*q is phi(p*q)/2. For phi see A000010.

Note: The order of 2 mod p*q is the smallest positive integer k such that 2^k = 1 mod p*q. See Rosen's definition of the order of an integer on p.334. Also, k is smaller than or equal to phi(p*q)/2 for all products of distinct odd primes p and q. See Cohen's Prop. 1.4.2 on p. 25.

2^(phi(p*q)/2) == 1 (mod p*q) for all distinct odd primes p and q. See Nagell's corollary to Theorem 64, p. 106, with a = 2 and n = p*q. - Wolfdieter Lang, Mar 31 2016

References

• Henri Cohen, Graduate Texts In Mathematics: A Course in Computational Algebraic Number Theory, Springer, 2000, p. 25
• Trygve Nagell, Introduction to Number Theory, Chelsea, 1964, p. 106.
• Kenneth H. Rosen, Elementary Number Theory And Its Applications, AT&T Laboratories, 2005, p. 334

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A005385, A007522.

Programs

Magma

[ p: p in PrimesUpTo(8000) | IsPrime((p-1) div 2) and not p mod 8 eq 7]; // Vincenzo Librandi, Feb 28 2016

Mathematica

Select[Prime@ Range@ 1000, And[PrimeQ[(# - 1)/2], MemberQ[Range[0, 6], Mod[#, 8]]] &] (* Michael De Vlieger, Feb 28 2016 *)

PARI

lista(nn) = {forprime(p=3, nn, if (((p % 8) != 7) && isprime((p-1)/2), print1(p, ", ")););} \\ Michel Marcus, Mar 24 2016

Formula

A005385 without its intersection with A007522.

Extensions

More terms from Vincenzo Librandi, Feb 28 2016