Kummer's theorem

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In mathematics, Kummer's theorem on binomial coeffients gives the highest power of a prime number p dividing a binomial coefficient. In particular, it asserts that given integers n ≥ m ≥ 0 and a prime number p, the maximum integer k such that pk divides the binomial coefficient \tbinom n m is equal to the number of carries when m is added to n − m in base p.

The theorem is named after Ernst Kummer, who proved it in the paper Kummer (1852).

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  • Kummer, Ernst (1852). "Über die Ergänzungssätze zu den allgemeinen Reciprocitätsgesetzen". Journal für die reine und angewandte Mathematik 44: 93–146.