A269454 - OEIS

%I #54 Sep 08 2022 08:46:15

%S 5,11,59,83,107,179,227,347,467,563,587,1019,1187,1283,1307,1523,1619,

%T 1907,2027,2099,2459,2579,2819,2963,3203,3467,3779,3803,3947,4139,

%U 4259,4283,4547,4787,5099,5387,5483,5507,5939,6659,6779,6827,6899,7187,7523

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

%C For safe primes see A005385.

%C 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.

%C 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.

%C 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

%D Henri Cohen, Graduate Texts In Mathematics: A Course in Computational Algebraic Number Theory, Springer, 2000, p. 25

%D Trygve Nagell, Introduction to Number Theory, Chelsea, 1964, p. 106.

%D Kenneth H. Rosen, Elementary Number Theory And Its Applications, AT&T Laboratories, 2005, p. 334

%H Amiram Eldar, <a href="/A269454/b269454.txt">Table of n, a(n) for n = 1..10000</a>

%F A005385 without its intersection with A007522.

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

%o (Magma) [ p: p in PrimesUpTo(8000) | IsPrime((p-1) div 2) and not p mod 8 eq 7]; // _Vincenzo Librandi_, Feb 28 2016

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

%Y Cf. A005385, A007522.

%K nonn

%O 1,1

%A _Marina Ibrishimova_, Feb 27 2016

%E More terms from _Vincenzo Librandi_, Feb 28 2016