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%I #25 Sep 08 2022 08:45:02
%S 149,223,593,1259,1481,1777,1999,2221,2591,2887,3109,3257,3331,3701,
%T 3923,4219,4441,4663,5107,5477,6143,6217,6661,6883,7253,7549,7919,
%U 7993,8363,8807,9029,9103,9473,9547,9769,10139,10657,11027,12211,12433
%N Primes p such that x^37 = 2 has no solution mod p.
%C Complement of A049569 relative to A000040.
%C Presumably this is also Primes congruent to 1 mod 37 (A216970). - _N. J. A. Sloane_, Jul 11 2008
%C 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
%H Vincenzo Librandi, <a href="/A059223/b059223.txt">Table of n, a(n) for n = 1..1000</a>
%t Select[Prime[Range[PrimePi[12500]]], ! MemberQ[PowerMod[Range[#], 37, #], Mod[2, #]] &] (* _T. D. Noe_, Sep 12 2012 *)
%t ok[p_]:= Reduce[Mod[x^37 - 2, p] == 0, x, Integers] == False; Select[Prime[Range[2000]], ok] (* _Vincenzo Librandi_, Sep 19 2012 *)
%o (Magma) [p: p in PrimesUpTo(13000) | forall{x: x in ResidueClassRing(p) | x^37 ne 2}]; // _Bruno Berselli_, Sep 12 2012
%o (PARI)
%o N=10^5; default(primelimit,N);
%o ok(p, r, k)={ return ( (p==r) || (Mod(r,p)^((p-1)/gcd(k,p-1))==1) ); }
%o forprime(p=2,N, if (! ok(p,2,37),print1(p,", ")));
%o /* _Joerg Arndt_, Sep 21 2012 */
%Y Cf. A000040, A049569, A216970.
%K nonn,easy
%O 1,1
%A _Klaus Brockhaus_, Jan 19 2001
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