

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).


4



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
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OFFSET

1,1


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 padic 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


PROG

(PARI) isA068209(n) = if(isprime(n) && n%3==2, for(a=1, n2, if(Mod(a+1, n^2)^n  Mod(a, n^2)^n==1, return(1)))); return(0) \\ Jianing Song, Nov 08 2022


CROSSREFS

Cf. A001220, A320535.
Sequence in context: A347804 A304356 A283146 * A139958 A097459 A145291
Adjacent sequences: A068206 A068207 A068208 * A068210 A068211 A068212


KEYWORD

easy,nonn


AUTHOR

Benoit Cloitre, Mar 23 2002


EXTENSIONS

Definition corrected by Mike Oakes, Feb 12 2009


STATUS

approved



