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

Author

Benoit Cloitre, Mar 23 2002

Keywords

• easy
• nonn

Comments

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)

Links

• 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

Crossrefs

Cf. A001220, A320535.

Programs

PARI

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

Extensions

Definition corrected by Mike Oakes, Feb 12 2009