Kummer congruences for the values b^r E_r(a/b) of the Euler polynomials at rational points, and

Abstract

Let E_r(x) be the Euler polynomials and let p be an odd prime. We prove that for every rational a/b with p not dividing b the numbers f_r = b^r E_r(a/b) are p-integral and satisfy f_r == f_r' (mod p^k) whenever r == r' (mod (p-1) p^(k-1)) and r, r' >= k; this is the only hypothesis. The proof is elementary: f_r is the p-adic limit of the alternating power sums Sum_{j < p^M} (-1)^j (bj + a)^r. The cases a/b = 0, 1/2, 1/4 give the Genocchi numbers G_(r+1)/(r+1), the signed Euler numbers and the signed Springer numbers, and hence "towers" a(n p^k) == ±a(n p^(k-1)) (mod p^k) for the tangent numbers A000182 (including P. Bala's shifted statement T(n p^k + i) == T(n p^(k-1) + i), the third statement of his comment of December 2021, which we prove for every prime, p = 2 by a valuation argument), the reduced tangent numbers A002105, the Euler numbers A000364, and, with a sign (-1)^(n p^(k-1)(p-1)/2), the zigzag numbers A000111 and the Springer numbers A001586. The sign law explains why the plain congruence holds for these two sequences exactly when p == 1 (mod 4), and for p == 3 (mod 4) exactly for even n; the exponent k is sharp in a majority of cases. Version 2 (October 16, 2026) adds Remark 4.1: all five towers hold in fact modulo p^(k + v_p(n)) for odd p, with the same sign, by applying the theorem to the pair (n_0, k + v_p(n)) where n = n_0 p^(v_p(n)). The refined exponent is attained in the verification range (p <= 31, k <= 5, n p^k <= 240) for every (p, k) with at least two admissible n, for the zigzag, Springer, Euler and reduced tangent numbers, and for the tangent numbers for p >= 5 (at p = 3 the differences have higher 3-adic valuation, an observation not pursued here). This refinement was found automatically by the exponent-form fitting of the author's SyntheticMind engine after version 1 was written; it is not in the journal version submitted on October 15, 2026. The verification script now checks it (section 5 of its output). No other changes. All statements except Bala's were produced by the generalization step of the SyntheticMind engine, which tests a checked congruence on related and classical sequences and returns those that pass with their sharp exponent; the sweep over the Euler-Bernoulli family found the plain tower for the zigzag and Springer numbers exactly at p = 5, 13, 17, 29 among p <= 31, which suggested the sign law. The proofs use no computation. Independent checks (Python 3, exact arithmetic: the general theorem on 72,258 pairs at nine rational points, the five towers on all odd primes p <= 31 with k <= 5, the sign controls, Bala's shifted statement on 2,637 cases for all primes p <= 31, and the refined exponent of version 2) are distributed with the note, and the text was reviewed by an independent AI referee with its own programs, which reproduced every count. The use of AI is described in the paper.