Connection between Fermat quotients and Euler numbers

Abstract

ABSTRACT. In the paper, we obtain a congruence modulo p 3 among Euler numbers E p−1 , E 2p−2 , and Fermat quotients Q 2 , Q a where p = a 2 + 4b 2 .

On some new congruences for generalized Bernoulli numbers

“…For congruences proved recently in [1], [2], [14], [17] or [18] we can often give much shorter proofs. Only Jakubec's congruence [8] seems to resist our methods so far.…”

“…Main result. The idea exploited in [1] to use identity (8) to extend classical congruences for the sums T r,k (n) seems to be very efficient. Identity (8) allows us to obtain almost automatically many new interesting congruences of Lerch [12], Lehmer [10] or Sun [17] types.…”

“…We shall use Cai's techniques to prove further congruences similar to those given in [10], [12] or [17]. For other results of the same type, not using (8), see [2] where almost all remaining congruences from [10] were extended.…”

“…Identity (8) was also used to show in an elementary way an extension of Gauss' congruence h(d) ≡ 0 (mod 2 ν−1 ), where ν is the number of distinct prime factors of the discriminant d of a quadratic field, to a similar congruence for generalized Bernoulli numbers, (B k,χ d /k) ≡ 0 (mod 2 ν−1 ). See [6] or [21].…”

Congruences related to Eulerian and Euler numbers

“…for p ≥ 5 (mod 3 l−1 ) for p = 3 Later, in [11], the authors extend these congruences to arbitrary moduli n. In an unrelated work in [22], by using some Galois theory, Stanislav Jakubec obtains a congruence modulo p 3 among the Euler numbers E p−1 and E 2p−2 and the Fermat quotients q 2 (p) and q a (p), where p = a 2 + 4b 2 is a prime such that p ≡ 1 mod 4. His congruence reads:…”