Two further towers of congruences: the Genocchi numbers with a factor p, and the denominators of the Bernoulli numbers

Abstract

Let G_m be the Genocchi numbers, G_m = 2(1 − 2^m) B_m, and p an odd prime. We prove that G_(2np^k) ≡ p · G_(2np^(k−1)) (mod p^(2k + 2 v_p(n))) for all n, k ≥ 1, and that this exponent is attained in most cases (552 of the 564 cases with p ≤ 59, k ≤ 3, 2np^k ≤ 800). The congruence is a direct consequence of the Kummer congruences for the values E_r(0) of the Euler polynomials proved in the author's previous note "Kummer congruences for the values of Euler polynomials at rational points, and towers of congruences for the tangent, Euler, zigzag and Springer numbers": since G_(2m) = 2m E_(2m−1)(0), the difference G_(2np^k) − p G_(2np^(k−1)) equals 2np^k (E_r(0) − E_r'(0)) with r ≡ r' (mod (p−1)p^(k−1)). The factor p explains a systematic observation: the plain tower G_(2np^k) ≡ G_(2np^(k−1)) (mod p^k) fails, and the minimum over n of v_p(difference) − k is exactly −1 for every (p, k). For p = 2 the statement is false (the twist c = 2 is not a 2-adic unit). We also record that the denominators D_(2n) of the Bernoulli numbers satisfy D_(2np^k) ≡ D_(2np^(k−1)) (mod p^k) for every prime p, with a one-line proof from the von Staudt–Clausen theorem: the primes q that divide D_(2np^k) but not D_(2np^(k−1)) satisfy q ≡ 1 (mod p^k). Both statements were found by a systematic exact search of the tower a(np^k) ≡ ±a(np^(k−1)) (mod p^E) over the Euler–Bernoulli family (zigzag, Springer, reduced tangent, Genocchi, Bernoulli numerators and denominators, tangent and Euler numbers), whose outcome is reported in the note; the search is now built into the author's SyntheticMind engine, which proposes the factor p and the sign automatically when a tower fails. The proofs were written with the assistance of an AI system under the author's direction; the author reviewed the content and is responsible for it.