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A358315 Primes p == 1 (mod 3) such that there exists 1 <= x <= p-2 such that (x+1)^p - x^p == 1 (mod p^2) and that p does not divide x^2 + x + 1.

%I A358315 #9 Nov 08 2022 18:25:11
%S A358315 79,193,337,421,457,547,601,619,691,757,787,907,1039,1093,1231,1237,
%T A358315 1303,1489,1531,1657,1993,2089,2113,2251,2311,2377,2389,2437,2539,
%U A358315 2647,2659,2713,2731,2749,3001,3037,3109,3229,3319,3331,3511,4003,4177,4243,4273,4339,4447
%N A358315 Primes p == 1 (mod 3) such that there exists 1 <= x <= p-2 such that (x+1)^p - x^p == 1 (mod p^2) and that p does not divide x^2 + x + 1.
%C A358315 If p == 1 (mod 3) and p divides x^2 + x + 1, then p^2 divides (x+1)^p - x^p - 1; see A068209 for a proof.
%C A358315 Primes p == 1 (mod 3) such that A320535(primepi(p)) > 2.
%C A358315 Conjecture: this density of this sequence among the primes congruent to 1 modulo 3 is the same as that of A068209 among the primes congruent to 2 modulo 3. - _Jianing Song_, Nov 08 2022
%H A358315 Jianing Song, <a href="/A358315/b358315.txt">Table of n, a(n) for n = 1..1312</a> (all terms up to 2*10^5)
%e A358315 For p = 79, the nontrivial solutions to (x+1)^p - x^p == 1 (mod p^2) are x == 11, 23, 32, 36, 42, 46, 55, 67 (mod 79). The equivalent classes x == 11, 32, 36, 42, 46, 67 (mod 79) satisfy x^2 + x + 1 != 0 (mod 79), so 79 is a term.
%o A358315 (PARI) isA358315(n) = if(isprime(n) && n%3==1, for(a=1, n-2, if(Mod(a+1,n^2)^n - Mod(a,n^2)^n==1 && znorder(Mod(a,n))!=3, return(1)))); return(0)
%Y A358315 Cf. A068209, A320535.
%K A358315 nonn
%O A358315 1,1
%A A358315 _Jianing Song_, Nov 08 2022