A059223 Primes p such that x^37 = 2 has no solution mod p.

149, 223, 593, 1259, 1481, 1777, 1999, 2221, 2591, 2887, 3109, 3257, 3331, 3701, 3923, 4219, 4441, 4663, 5107, 5477, 6143, 6217, 6661, 6883, 7253, 7549, 7919, 7993, 8363, 8807, 9029, 9103, 9473, 9547, 9769, 10139, 10657, 11027, 12211, 12433

Offset: 1

Author

Klaus Brockhaus, Jan 19 2001

Keywords

• nonn
• easy

Comments

Complement of A049569 relative to A000040.

Presumably this is also Primes congruent to 1 mod 37 (A216970). - N. J. A. Sloane, Jul 11 2008

Not so. The smallest counterexample is 11471: 11471 == 1 (mod 37), but 43^37 == 2 (mod 11471), 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, A049569, A216970.

Programs

Magma

[p: p in PrimesUpTo(13000) | forall{x: x in ResidueClassRing(p) | x^37 ne 2}]; // Bruno Berselli, Sep 12 2012

Mathematica

Select[Prime[Range[PrimePi[12500]]], ! MemberQ[PowerMod[Range[#], 37, #], Mod[2, #]] &] (* T. D. Noe, Sep 12 2012 *)
ok[p_]:= Reduce[Mod[x^37 - 2, p] == 0, x, Integers] == False; Select[Prime[Range[2000]], ok] (* Vincenzo Librandi, Sep 19 2012  *)

PARI

N=10^5;  default(primelimit,N);
ok(p, r, k)={ return ( (p==r) || (Mod(r,p)^((p-1)/gcd(k,p-1))==1) ); }
forprime(p=2,N, if (! ok(p,2,37),print1(p,", ")));
/* Joerg Arndt, Sep 21 2012 */