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.

79, 193, 337, 421, 457, 547, 601, 619, 691, 757, 787, 907, 1039, 1093, 1231, 1237, 1303, 1489, 1531, 1657, 1993, 2089, 2113, 2251, 2311, 2377, 2389, 2437, 2539, 2647, 2659, 2713, 2731, 2749, 3001, 3037, 3109, 3229, 3319, 3331, 3511, 4003, 4177, 4243, 4273, 4339, 4447

Offset: 1

Author

Jianing Song, Nov 08 2022

Keywords

• nonn

Comments

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.

Primes p == 1 (mod 3) such that A320535(primepi(p)) > 2.

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

Examples

			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.
		

Links

• Jianing Song, Table of n, a(n) for n = 1..1312 (all terms up to 2*10^5)

Crossrefs

Cf. A068209, A320535.

Programs

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)