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

631, 673, 953, 1163, 1709, 2003, 2143, 2731, 2857, 3109, 3389, 3739, 4271, 4999, 5237, 5279, 5531, 5867, 6553, 6679, 6959, 7001, 7309, 7351, 7393, 8191, 8681, 9157, 9829, 10627, 10739, 11117, 11243, 11299, 11411, 11467, 13007, 13259, 15121, 15233, 15583, 16073, 18439, 18803, 20063, 20147

Offset: 1

Author

N. J. A. Sloane, Apr 01 2016

Keywords

• nonn

Links

• Robert Israel, Table of n, a(n) for n = 1..10000
• Leonard Eugene Dickson, Cyclotomy and trinomial congruences, Transactions of the American Mathematical Society, 37.3 (1935): 363-380. See page 373.

Crossrefs

Cf. A042966.

Programs

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

Maple

ans:=[];
M:=10000;
e:=7; r:=2;
for k from 2 to M do
    p:=ithprime(k);
    if p mod 14 = 1 then
       for x from 2 to p-1 do
          if x^e mod p = r then
             ans:=[op(ans),p];
             break;
          end if;
       end do:
    end if;
end do:
ans;
# Alternative:
select(p -> isprime(p) and numtheory:-mroot(2,7,p)<>FAIL, [seq(14*i+1,i=1..3000)]); # Robert Israel, Apr 03 2018

Mathematica

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 *)

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