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Thursday, December 6, 2018

**Abstract:** Abstract:That $(n+1)| binom{2n}{n}$ for every positive integer $n$ is a consequence of the integrality of the Catalan numbers. Pomerance conjectured that $n| binom{2n}{n}$ for a set of positive density. In joint work with Sergei Konyagin, we prove Pomerance's conjecture and give an explicit form of the density. The proof uses a new method for accounting for the largest prime factors of integers.