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 (mod p α) pn + s pk + t (−1) pk k − r p α−1 l ≡ 1 ⌊n/p α−1 ⌋! k≡r (mod p α) n k s t (−1) k k − r p α−1 l (mod p). We also present an application of the first congruence to Bernoulli polyno-mials, and apply the second congruence to show that a p-adic order bound given by the authors in a previous paper is sharp.