Resumen de Combinatorial congruences modulo prime powers

Zhi-Wei Sun, Donald M. Davis

• Let be any prime, and let and be nonnegative integers. Let and . We establish the congruence (motivated by a conjecture arising from algebraic topology) and obtain the following vast generalization of Lucas' theorem: If is greater than one, and are nonnegative integers with , then We also present an application of the first congruence to Bernoulli polynomials and apply the second congruence to show that a -adic order bound given by the authors in a previous paper can be attained when .