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

1,1

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

Mathematica

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

Prog

(Magma) [p: p in PrimesUpTo(500) | not exists{x : x in ResidueClassRing(p) | x^8 eq 2} ]; // 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

Crossrefs

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

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

Sequence in context: A059641 A014662 A059349 * A040100 A076757 A178942

Adjacent sequences: A045313 A045314 A045315 * A045317 A045318 A045319

Keyword

nonn,easy

Author

N. J. A. Sloane

Status

approved