

A269454


Safe primes that are not congruent to 1 mod 8.


4



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
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OFFSET

1,1


COMMENTS

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



FORMULA



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((p1) 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((p1)/2), print1(p, ", ")); ); } \\ Michel Marcus, Mar 24 2016


CROSSREFS



KEYWORD

nonn


AUTHOR



EXTENSIONS



STATUS

approved



