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

%I A059236 #35 Sep 08 2022 08:45:02
%S A059236 83,739,821,1231,1559,1723,2297,2543,2707,2789,2953,3527,3691,4019,
%T A059236 5003,5167,5413,5659,5741,5987,6151,6397,6971,7873,8447,8693,9103,
%U A059236 9349,9431,9677,9923,10169,10333,11071,11317,11399,12301,12547,13121,13367
%N A059236 Primes p such that x^41 = 2 has no solution mod p.
%C A059236 Complement of A049573 relative to A000040.
%C A059236 Presumably this is also "Primes congruent to 1 mod 41" (A212379), but that requires a proof. - _N. J. A. Sloane_, Jul 11 2008
%C A059236 Smallest counterexample: 17467 is not in A059236, but congruent to 1 mod 41 (17467 = 426*41+1). - _Klaus Brockhaus_, May 18 2011
%H A059236 Klaus Brockhaus, <a href="/A059236/b059236.txt">Table of n, a(n) for n = 1..100000</a>
%t A059236 ok[p_]:= Reduce[Mod[x^41 - 2, p] == 0, x, Integers] == False; Select[Prime[Range[2000]], ok] (* _Vincenzo Librandi_, Sep 20 2012 *)
%t A059236 Select[Prime[Range[PrimePi[14000]]], ! MemberQ[PowerMod[Range[#], 41, #], Mod[2, #]] &] (* _Bruno Berselli_, Sep 20 2012 *)
%o A059236 (Magma) [p: p in PrimesUpTo(13400) | not exists{x: x in ResidueClassRing(p) | x^41 eq 2}]; // _Klaus Brockhaus_, May 18 2011
%o A059236 (Magma) /* Alternatively: */ [p: p in PrimesUpTo(13400) | forall{x: x in ResidueClassRing(p) | x^41 ne 2}]; // _Bruno Berselli_, Sep 20 2012
%o A059236 (PARI) forprime(p=2, 69589957, if(trap(, 1, sqrtn(Mod(2, p), 41); 0), print1(p,", "))) \\ _Klaus Brockhaus_, May 18 2011
%o A059236 (PARI)
%o A059236 N=10^5;  default(primelimit,N);
%o A059236 ok(p, r, k)={ return ( (p==r) || (Mod(r,p)^((p-1)/gcd(k,p-1))==1) ); }
%o A059236 forprime(p=2,N, if (! ok(p,2,41),print1(p,", ")));
%o A059236 /* _Joerg Arndt_, Sep 21 2012 */
%Y A059236 Subsequence of A212379.
%Y A059236 Cf. A049573, A142199, A142200, A190758.
%K A059236 nonn,easy
%O A059236 1,1
%A A059236 _Klaus Brockhaus_, Jan 20 2001