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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. 1

%I

%S 631,673,953,1163,1709,2003,2143,2731,2857,3109,3389,3739,4271,4999,

%T 5237,5279,5531,5867,6553,6679,6959,7001,7309,7351,7393,8191,8681,

%U 9157,9829,10627,10739,11117,11243,11299,11411,11467,13007,13259,15121,15233,15583,16073,18439,18803,20063,20147

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

%H Robert Israel, <a href="/A270802/b270802.txt">Table of n, a(n) for n = 1..10000</a>

%H 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 ans:=[];

%p M:=10000;

%p e:=7; r:=2;

%p for k from 2 to M do

%p p:=ithprime(k);

%p if p mod 14 = 1 then

%p for x from 2 to p-1 do

%p if x^e mod p = r then

%p ans:=[op(ans),p];

%p break;

%p end if;

%p end do:

%p end if;

%p end do:

%p ans;

%p # Alternative:

%p select(p -> isprime(p) and numtheory:-mroot(2,7,p)<>FAIL, [seq(14*i+1,i=1..3000)]); # _Robert Israel_, Apr 03 2018

%t 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 (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 (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 Cf. A042966.

%K nonn

%O 1,1

%A _N. J. A. Sloane_, Apr 01 2016

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Last modified February 17 12:32 EST 2020. Contains 331996 sequences. (Running on oeis4.)