A059245 Primes p such that x^13 = 2 has no solution mod p.

53, 79, 131, 157, 313, 443, 521, 547, 599, 677, 859, 911, 937, 1093, 1171, 1223, 1249, 1301, 1327, 1483, 1613, 1847, 1873, 1951, 2003, 2029, 2081, 2237, 2341, 2393, 2549, 2731, 2861, 2887, 2939, 3121, 3251, 3329, 3407, 3433, 3511, 3719, 3797, 3823, 4057

Offset: 1

Author

Klaus Brockhaus, Jan 21 2001

Keywords

• nonn
• easy

Comments

Complement of A049545 relative to A000040.

Presumably this is the same as Primes congruent to 1 mod 13. - N. J. A. Sloane, Jul 11 2008

The smallest counterexample is 4421: 4421 == 1 (mod 13), but 162^13 == 2 (mod 4421), therefore this prime is not in the sequence. - Bruno Berselli, Sep 12 2012

Links

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

Crossrefs

Cf. A000040, A049545, A268753.

Programs

Magma

[p: p in PrimesUpTo(4500) | forall{x: x in ResidueClassRing(p) | x^13 ne 2}]; // Bruno Berselli, Sep 12 2012

Mathematica

Select[Prime[Range[PrimePi[5000]]], ! MemberQ[PowerMod[Range[#], 13, #], Mod[2, #]] &] (* T. D. Noe, Sep 12 2012 *)
ok[p_]:= Reduce[Mod[x^13 - 2, p] == 0, x, Integers] == False; Select[Prime[Range[600]], ok] (* Vincenzo Librandi, Sep 20 2012  *)