Mathematics Colloquium: Prime Divisors of Binomial Coefficients

2017-11-09

Neill 5W, 4:10pm

Matthew Hudelson

We examine some results that center around prime divisors of binomial coefficients, leading off with Lucas’ theorem—a result that greatly facilitates computing (𝑛/k) modulo p. This result has a striking visual connection with the Sierpinski triangle when p=2. As a follow up from Lucas’ theorem, we will examine various quick results concerning what prime powers pk divide binomial coefficients. The second result we will examine is Bertrand’s postulate: For any integer n ≥ 1, there is always a prime number in [n, 2n]. We will present a version of Erdős’ proof of Bertrand’s postulate that uses remarkably crude estimates concerning primes that divide the central binomial coefficient (2𝑛/𝑛). Refreshments served during the event!