Universal Kummer congruences mod prime powers

Universal Kummer Congruences Mod Prime Powers

The Universal Kummer Congruences

Let p be a prime. In this paper, we present detailed p-adic analysis to factorials and double factorials and their congruences. We give good bounds for the p-adic sizes of the coefficients of the divided universal Bernoulli number B̂n n when n is divisible by p−1. Using these we then establish the universal Kummer congruences modulo powers of a prime p for the divided universal Bernoulli numbers...

Congruences modulo Prime Powers

Let p be any prime, and let α and n be nonnegative integers. Let r ∈ Z and f (x) ∈ Z[x]. We establish the congruence p deg f k≡r (mod p α) n k (−1) k f k − r p α ≡ 0 mod p ∞ i=α ⌊n/p i ⌋ (motivated by a conjecture arising from algebraic topology), and obtain the following vast generalization of Lucas' theorem: If α > 1 and l, s, t are nonnegative integers with s, t < p, then 1 ⌊n/p α−1 ⌋! k≡r (...

Combinatorial Congruences modulo Prime Powers

Let p be any prime, and let α and n be nonnegative integers. Let r ∈ Z and f(x) ∈ Z[x]. We establish the congruence p f ∑ k≡r (mod pα) (n k ) (−1)f ( k − r pα ) ≡ 0 ( mod p ∑∞ i=α n/p i ) (motivated by a conjecture arising from algebraic topology) and obtain the following vast generalization of Lucas’ theorem: If α is greater than one, and l, s, t are nonnegative integers with s, t < p, then 1 ...

Bounds of Divided Universal Bernoulli Numbers and Universal Kummer Congruences

Let p be a prime. We obtain good bounds for the p-adic sizes of the coefficients of the divided universal Bernoulli number B̂n n when n is divisible by p− 1. As an application, we give a simple proof of Clarke’s 1989 universal von Staudt theorem. We also establish the universal Kummer congruences modulo p for the divided universal Bernoulli numbers for the case (p − 1)|n, which is a new result.