Mathematics Colloquium: Prime Divisors of Binomial Coefficients
Neill 5W, 4:10pm
We examine some results that center around prime divisors of binomial coefficients, leading off with Lucasâ€™ theoremâ€”a result that greatly facilitates computing (ð‘›/k) modulo p. This result has a striking visual connection with the Sierpinski triangle when p=2. As a follow up from Lucasâ€™ theorem, we will examine various quick results concerning what prime powers pk divide binomial coefficients. The second result we will examine is Bertrandâ€™s postulate: For any integer n â‰¥ 1, there is always a prime number in [n, 2n]. We will present a version of ErdÅ‘sâ€™ proof of Bertrandâ€™s postulate that uses remarkably crude estimates concerning primes that divide the central binomial coefficient (2ð‘›/ð‘›). Refreshments served during the event!