Let p be a composite, and let us apply the simple rule stated by Lucas' theorem to check whether p is prime (that is, involving the p base representation). Now, assume that the result is positive; would that mean that indeed p divides $n \choose k$, or the result is random, since the hypothesis of p being a prime is violated?
2026-03-27 14:57:44.1774623464
On Lucas' Theorem over composite
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