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A270802 Primes p of the form 14*k+1 for which there is a solution to x^7 == 2 mod p.

%I A270802 #38 Sep 08 2022 08:46:16
%S A270802 631,673,953,1163,1709,2003,2143,2731,2857,3109,3389,3739,4271,4999,
%T A270802 5237,5279,5531,5867,6553,6679,6959,7001,7309,7351,7393,8191,8681,
%U A270802 9157,9829,10627,10739,11117,11243,11299,11411,11467,13007,13259,15121,15233,15583,16073,18439,18803,20063,20147
%N A270802 Primes p of the form 14*k+1 for which there is a solution to x^7 == 2 mod p.
%H A270802 Robert Israel, <a href="/A270802/b270802.txt">Table of n, a(n) for n = 1..10000</a>
%H A270802 Leonard Eugene Dickson, <a href="http://dx.doi.org/10.1090/S0002-9947-1935-1501791-3">Cyclotomy and trinomial congruences</a>, Transactions of the American Mathematical Society, 37.3 (1935): 363-380. See page 373.
%p A270802 ans:=[];
%p A270802 M:=10000;
%p A270802 e:=7; r:=2;
%p A270802 for k from 2 to M do
%p A270802     p:=ithprime(k);
%p A270802     if p mod 14 = 1 then
%p A270802        for x from 2 to p-1 do
%p A270802           if x^e mod p = r then
%p A270802              ans:=[op(ans),p];
%p A270802              break;
%p A270802           end if;
%p A270802        end do:
%p A270802     end if;
%p A270802 end do:
%p A270802 ans;
%p A270802 # Alternative:
%p A270802 select(p -> isprime(p) and numtheory:-mroot(2,7,p)<>FAIL, [seq(14*i+1,i=1..3000)]); # _Robert Israel_, Apr 03 2018
%t A270802 Select[Select[14 Range[10^3] + 1, PrimeQ], Function[p, AnyTrue[Range[2, 10^4], Mod[#^7, p] == 2 &]]] (* _Michael De Vlieger_, Apr 02 2016, Version 10 *)
%o A270802 (Magma) [p: p in PrimesUpTo(50000) | IsOne(p mod 14) and exists{x: x in ResidueClassRing(p) | x^7 eq 2}]; // _Bruno Berselli_, Apr 02 2016
%o A270802 (PARI) forprime(p=2,10^5,if(p%14!=1,next);if(Mod(2,p)^((p-1)/7)==1,print1(p,", "))); \\ _Joerg Arndt_, Apr 03 2016
%Y A270802 Cf. A042966.
%K A270802 nonn
%O A270802 1,1
%A A270802 _N. J. A. Sloane_, Apr 01 2016