Powers of two weighted sum of the first p divided Bernoulli numbers modulo p

Abstract

We show that, modulo some odd prime p, the powers of two weighted sum of the first p divided Bernoulli numbers equals twice the number of permutations on p − 2 letters with an even number of ascents and distinct from the identity. We provide a combinatorial characterization of Wieferich primes, as well as of primes p for which p 2 divides the Fermat quotient q p (2).

Congruences related to Eulerian and Euler numbers

“…It is shown in [27] that the sum of the first p−1 2 odd powers of the first p−1 2 integers is congruent to − 1 2 modulo p. This is Theorem 7 of [27]. Expressed in terms of the S k 's, we thus have:…”

“…The last congruence holds as by the Von Staudt-Clausen's theorem [42][9], we have pB p−1 = −1 mod p. It then suffices to apply Lemma 3 above and Theorem 1 of [27] in order to conclude.…”

“…The proof of Lemma 1 relies on Identity ( 5) in which we use Fermat's little theorem and the fact that for this whole range of s, we have (see Lemma 5 of [27]):…”

“…Proof of Lemma 2. The proof relies on a result of [27] providing the residue of the Fermat quotient q 2 (p) in terms of the residue of twice the number of permutations of Sym(p − 2) with an even number of ascents. A reformulation of Theorem 1 of [27] is:…”

“…Identity ( 27) and ( 28) are respectively Theorem 2 and 4 of [36]. In Identity (27), it is understood that the binomial transforms are zero when the integer is negative. Identity ( 24) is simply obtained by using (28) Their result from [34] is based in particular on Zeilberger's algorithm for definite hypergeometric sums, see [49], and on expressing harmonic numbers in terms of differentiation of binomial coefficients, a method which can be traced back to Isaac Newton.…”