Emmy-Noether-Seminar

Datum: 03-05-2019Zeit: 13:00 – 14:00Ort: Raum 04.363, Cauerstraße 11, Erlangen

Binomial divisibility out to 128 billion

Russ Woodroofe (Pristina)

Abstract: 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 it 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 128 billion. This computation finishes in about 50 hours, which is a great improvement over the 2 weeks that our earlier computation out to 1 billion took.

In this talk, I'll tell you how the question arose in our work. I'll then overview the interplay between group theory, number theory, and computation allowing this computation.

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Details

Datum:
03-05-2019
Zeit:
13:00 – 14:00
Ort:

Raum 04.363, Cauerstraße 11, Erlangen

Veranstaltungskategorien:
Emmy Noether