Kummer's Theorem

In this video, I go over the proofs of Legendre's formula and Kummer's theorem to establish a useful corollary for some upcoming Hilbert's Tenth Problem videos: the sum of the digits in binary of a number n is the exponent of 2 in the prime factorization of 2n choose n. Kummer's theorem states that the exponent of p in the prime factorization of n+m choose n is the number of carries when adding n and m in base p. Proofs are from Wikipedia: https://en.wikipedia.org/wiki/Kummer%... https://en.wikipedia.org/wiki/Legendr...