A059236 Primes p such that x^41 = 2 has no solution mod p.

83, 739, 821, 1231, 1559, 1723, 2297, 2543, 2707, 2789, 2953, 3527, 3691, 4019, 5003, 5167, 5413, 5659, 5741, 5987, 6151, 6397, 6971, 7873, 8447, 8693, 9103, 9349, 9431, 9677, 9923, 10169, 10333, 11071, 11317, 11399, 12301, 12547, 13121, 13367

Offset: 1

Author

Klaus Brockhaus, Jan 20 2001

Keywords

• nonn
• easy

Comments

Complement of A049573 relative to A000040.

Presumably this is also "Primes congruent to 1 mod 41" (A212379), but that requires a proof. - N. J. A. Sloane, Jul 11 2008

Smallest counterexample: 17467 is not in A059236, but congruent to 1 mod 41 (17467 = 426*41+1). - Klaus Brockhaus, May 18 2011

Links

• Klaus Brockhaus, Table of n, a(n) for n = 1..100000

Crossrefs

Subsequence of A212379.

Cf. A049573, A142199, A142200, A190758.

Programs

Magma

[p: p in PrimesUpTo(13400) | not exists{x: x in ResidueClassRing(p) | x^41 eq 2}]; // Klaus Brockhaus, May 18 2011

Magma

/* Alternatively: */ [p: p in PrimesUpTo(13400) | forall{x: x in ResidueClassRing(p) | x^41 ne 2}]; // Bruno Berselli, Sep 20 2012

Mathematica

ok[p_]:= Reduce[Mod[x^41 - 2, p] == 0, x, Integers] == False; Select[Prime[Range[2000]], ok] (* Vincenzo Librandi, Sep 20 2012 *)
Select[Prime[Range[PrimePi[14000]]], ! MemberQ[PowerMod[Range[#], 41, #], Mod[2, #]] &] (* Bruno Berselli, Sep 20 2012 *)

PARI

forprime(p=2, 69589957, if(trap(, 1, sqrtn(Mod(2, p), 41); 0), print1(p,", "))) \\ Klaus Brockhaus, May 18 2011

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,2,41),print1(p,", ")));
/* Joerg Arndt, Sep 21 2012 */