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

1,1

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

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 *)

Prog

(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
(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 */

Crossrefs

Subsequence of A212379.

Cf. A049573, A142199, A142200, A190758.

Sequence in context: A164758 A142751 A176633 * A212379 A059935 A069596

Adjacent sequences: A059233 A059234 A059235 * A059237 A059238 A059239

Keyword

nonn,easy

Author

Klaus Brockhaus, Jan 20 2001

Status

approved