Let p be any prime, and let a and n be nonnegative integers. Let $r\in Z$ and $f(x)\in Z[x]$. We establish the congruence $$p^{ \deg f}\sum_{ k=r(mod p^a)}\binom{ n}{ k}(-1)^k f((k-r)/p^a) =0 (mod p^{ \sum_{ i=a}^{ \infty}[n/p^i]})$$ (motivated by a conjecture arising from algebraic topology), and obtain the following vast generalization of Lucas' theorem: If a is greater than one, and $l,s,t$ are nonnegative integers with $s,t<p$, then $$\frac{ 1}{ [n/p^{ a-1}]!} \sum_{ k=r(mod p^a)} \binom{ pn+s}{ pk+t}(-1)^{ pk}((k-r)/p^{ a-1})^l =\frac { 1}{ [n/p^{ a-1}]!} \sum_{ k=r(mod p^a)}\binom{ n}{ k}\binom{ s}{ t}(-1)^k((k-r)/p^{ a-1})^l (mod p).$$ We also present an application of the first congruence to Bernoulli polynomials, and apply the second congruence to show that a p-adic order bound given by the authors in a previous paper can be attained when p=2.
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