cp's OEIS Frontend

A097955 Primes p such that p divides 5^((p-1)/2) - 2^((p-1)/2).

%I A097955 #38 Oct 13 2022 13:57:03
%S A097955 3,13,31,37,41,43,53,67,71,79,83,89,107,151,157,163,173,191,197,199,
%T A097955 227,239,241,271,277,281,283,293,307,311,317,347,359,373,397,401,409,
%U A097955 431,439,443,449,467,479,521,523,547,557,563,569,587,599,601,613,631,641
%N A097955 Primes p such that p divides 5^((p-1)/2) - 2^((p-1)/2).
%C A097955 Also 3 and primes p such that (p^2 - 1)/24 mod 10 = {0, 7}. - _Richard R. Forberg_, Aug 31 2013
%C A097955 Also primes p such that x^2 = 10 mod p has integer solutions, or Legendre(10, p) = 1. However, p could be irreducible but not prime in Z[sqrt(10)], especially if p = 3 or 7 mod 10. - _Alonso del Arte_, Dec 27 2015
%C A097955 Rational primes that decompose in the field Q(sqrt(10)). - _N. J. A. Sloane_, Dec 26 2017
%C A097955 From _Jianing Song_, Oct 13 2022: (Start)
%C A097955 Primes p such that kronecker(10,p) = 1 (or equivalently, kronecker(40,p) = 1).
%C A097955 Primes congruent to 1, 3, 9, 13, 27, 31, 37, 39 modulo 40. (End)
%H A097955 Robert Israel, <a href="/A097955/b097955.txt">Table of n, a(n) for n = 1..10000</a>
%H A097955 <a href="/index/Pri#primes_decomp_of">Index to sequences related to decomposition of primes in quadratic fields</a>
%e A097955 For p = 13, 5^6 - 2^6 = 15561 is divisible by 13, so 13 is in the sequence.
%p A097955 select(p -> isprime(p) and  10 &^ ((p-1)/2)  mod p = 1, [seq(i,i=3..1000,2)]); # _Robert Israel_, Dec 28 2015
%t A097955 Select[Prime[Range[100]], JacobiSymbol[10, #] == 1 &] (* _Alonso del Arte_, Dec 27 2015 *)
%o A097955 (PARI) \\ s = +-1,d=diff
%o A097955 ptopm1d2(n,x,d,s) = { forprime(p=3,n,p2=(p-1)/2; y=x^p2 + s*(x-d)^p2; if(y%p==0,print1(p","))) }
%o A097955 ptopm1d2(1000, 5, 3, -1)
%o A097955 (PARI) isA097955(p) == isprime(p) && kronecker(10,p) == 1 \\ _Jianing Song_, Oct 13 2022
%Y A097955 A038879, the sequence of primes that do not remain inert in the field Q(sqrt(10)), is essentially the same.
%Y A097955 Cf. A038880 (rational primes that remain inert in the field Q(sqrt(10))).
%K A097955 nonn,easy
%O A097955 1,1
%A A097955 _Cino Hilliard_, Sep 06 2004