A320535 -id:A320535 - cp's OEIS frontend

A068209 Primes p of the form 3k - 1 such that there exist nontrivial solutions (x other than 0 or -1 modulo p) to the congruence (x+1)^p - x^p == 1 (mod p^2).

59, 83, 179, 227, 419, 443, 701, 857, 887, 911, 929, 971, 977, 1091, 1109, 1193, 1217, 1223, 1259, 1283, 1289, 1439, 1487, 1493, 1613, 1637, 1811, 1847, 1901, 1997, 2003, 2087, 2243, 2423, 2477, 2579, 2591, 2729, 2777, 2969, 3089, 3137, 3191, 3203, 3251

Offset: 1

Benoit Cloitre, Mar 23 2002

Note that nontrivial solutions always exist for primes of the form 3k + 1. - Jianing Song, Apr 20 2019
From Jianing Song, Nov 08 2022: (Start)
Proof: for prime p > 3, write f_p(x) = ((x+1)^p - x^p - 1)/p; f_p(x) is a polynomial in Z[x]. We have f_p(w) = f_p(w^2) = 0, where w is a primitive cube root of 1, so f_p(x) divides x^2 + x + 1 in Q[x]. Since x^2 + x + 1 is a primitive polynomial (having coprime coefficients), it follows from Gauss's lemma for polynomials that f_p(x) divides x^2 + x + 1 in Z[x]. As a result, if p == 1 (mod 3) and p | (x^2 + x + 1) for some x, then p^2 | ((x+1)^p - x^p - 1).
For prime p > 2, the equation x^p + y^p = z^p has nontrivial solutions over (Z_p)* (the p-adic integers not divisible by p) if and only if there exist nontrivial solutions to the congruence (x+1)^p - x^p == 1 (mod p^2). (End)

Jianing Song, Table of n, a(n) for n = 1..1397 (all terms up to 2*10^5; first 74 terms from Robert G. Wilson v)
K. S. Brown, On the Density of Some Exceptional Primes

Cf. A001220, A320535.

isA068209(n) = if(isprime(n) && n%3==2, for(a=1, n-2, if(Mod(a+1,n^2)^n - Mod(a,n^2)^n==1, return(1)))); return(0) \\ Jianing Song, Nov 08 2022

Definition corrected by Mike Oakes, Feb 12 2009

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.

Original entry on oeis.org

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

Jianing Song, Nov 08 2022

nonn

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

			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.

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

Cf. A068209, A320535.

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)

cp's OEIS Frontend

A068209 Primes p of the form 3k - 1 such that there exist nontrivial solutions (x other than 0 or -1 modulo p) to the congruence (x+1)^p - x^p == 1 (mod p^2).

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