Binomial divisibility to 7 billion

In earlier work with John Shareshian, we asked whether for every number n, there are primes p and r so that every nontrivial binomial coefficient n choose k is divisible by at least one of the two primes.  The motivation for the question is from group theory: the question is equivalent to that of whether for every n there are primes p,r so that the alternating group A_n is generated by any Sylow subgroups at p and r.  The answer is "yes" up to a set of asymptotic density zero (assuming the Riemann Hypothesis), and we verified a "yes" answer computationally out to 1 billion.

In more recent work, joined by Bob Guralnick, we have verified computationally that a stronger condition holds for all n out to 7 billion.  This computation finishes in a few hours, which is a great improvement over the 2 weeks that our earlier computation out to 1 billion took.

In this talk, I'll overview the interplay between group theory, number theory, and computation allowing this computation.