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A097959 Primes p such that p divides 6^((p-1)/2) - 5^((p-1)/2).

%I A097959 #25 Oct 13 2022 13:57:11
%S A097959 7,13,17,19,29,37,71,83,101,103,107,113,127,137,139,149,157,191,211,
%T A097959 223,227,233,239,241,257,269,277,311,331,347,353,359,367,373,379,389,
%U A097959 397,409,431,443,461,463,467,479,487,499,509,563,571,587,593,599,601,607
%N A097959 Primes p such that p divides 6^((p-1)/2) - 5^((p-1)/2).
%C A097959 From _Jianing Song_, Oct 13 2022: (Start)
%C A097959 Rational primes that decompose in the field Q(sqrt(30)).
%C A097959 Primes p such that kronecker(30,p) = 1 (or equivalently, kronecker(120,p) = 1).
%C A097959 Primes congruent to 1, 7, 13, 17, 19, 29, 37, 49, 71, 83, 91, 101, 103, 107, 113, 119 modulo 120. (End)
%H A097959 Amiram Eldar, <a href="/A097959/b097959.txt">Table of n, a(n) for n = 1..10000</a>
%H A097959 <a href="/index/Pri#primes_decomp_of">Index to sequences related to decomposition of primes in quadratic fields</a>
%e A097959 7 is a term since it is a prime and 6^((7-1)/2) - 5^((7-1)/2) = 6^3 - 5^3 = 91 = 7*13 is divisible by 7.
%t A097959 Select[Prime[Range[200]],Divisible[6^((#-1)/2)-5^((#-1)/2),#]&] (* _Harvey P. Dale_, Jun 06 2018 *)
%t A097959 Select[Range[3, 600, 2], PrimeQ[#] && PowerMod[5, (# - 1)/2, #] == PowerMod[6, (# - 1)/2, #] &] (* _Amiram Eldar_, Apr 07 2021 *)
%o A097959 (PARI) \\ s = +-1, d=diff
%o A097959 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 A097959 ptopm1d2(1000, 6, 1, -1)
%o A097959 (PARI) isA097959(p) == isprime(p) && kronecker(30, p) == 1 \\ _Jianing Song_, Oct 13 2022
%Y A097959 A038903, the sequence of primes that do not remain inert in the field Q(sqrt(30)), is essentially the same.
%Y A097959 Cf. A038904 (rational primes that remain inert in the field Q(sqrt(30))).
%K A097959 nonn,easy
%O A097959 1,1
%A A097959 _Cino Hilliard_, Sep 06 2004
%E A097959 Definition clarified by _Harvey P. Dale_, Jun 06 2018