OFFSET
1,1
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
MATHEMATICA
Select[Prime@ Range@ 1000, And[PrimeQ[(# - 1)/2], MemberQ[Range[0, 6], Mod[#, 8]]] &] (* Michael De Vlieger, Feb 28 2016 *)
PROG
(Magma) [ p: p in PrimesUpTo(8000) | IsPrime((p-1) div 2) and not p mod 8 eq 7]; // Vincenzo Librandi, 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
CROSSREFS
KEYWORD
nonn
AUTHOR
Marina Ibrishimova, Feb 27 2016
EXTENSIONS
More terms from Vincenzo Librandi, Feb 28 2016
STATUS
approved