A045316 Primes p such that x^8 = 2 has no solution mod p.

3, 5, 11, 13, 17, 19, 29, 37, 41, 43, 53, 59, 61, 67, 83, 97, 101, 107, 109, 113, 131, 137, 139, 149, 157, 163, 173, 179, 181, 193, 197, 211, 227, 229, 241, 251, 269, 277, 281, 283, 293, 307, 313, 317, 331, 347, 349, 353, 373, 379, 389

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

Complement of A045315 relative to A000040. Coincides for the first 140 terms with the sequence of primes p such that x^16 = 2 has no solution mod p (first divergence is at 1217, cf. A059287). - Klaus Brockhaus, Jan 26 2001

Differs from A059349 (x^32 == 2 (mod p) has no solution) first at a(37) = A059349(38), the term A059349(37) = 257 which is not in this sequence. See A070184 for all such terms. - M. F. Hasler, Jun 21 2024

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A000040, A045315 (complement in the primes), A059287.

Subsequence of A059349 (same with x^32), complement is A070184.

Programs

Magma

[p: p in PrimesUpTo(500) | not exists{x : x in ResidueClassRing(p) | x^8 eq 2} ]; // Vincenzo Librandi, Sep 19 2012

Mathematica

ok[p_]:= Reduce[Mod[x^8 - 2, p] == 0, x, Integers] == False; Select[Prime[Range[200]], ok] (* Vincenzo Librandi, Sep 19 2012 *)

PARI

select( {is_A045316(p)=Mod(2,p)^(p\gcd(8,p-1))!=1 && p>2}, primes(199)) \\ Append "&& isprime(p)" if that's not known. - M. F. Hasler, Jun 22 2024