Eric Rowland, Hofstra University

Abstract

In the 1870s, Lucas obtained a beautiful formula for binomial coefficients modulo p. Namely, Binomial[n, m] is congruent modulo p to the product of the binomial coefficients whose arguments are the base-p digits of n and m, taken pairwise. Variations and generalizations of this formula have been actively investigated since. In particular, for which n and m does Lucas' congruence hold not just modulo p but modulo p2? The answer is related to some hidden rotational symmetry in Pascal's triangle. A similar result holds for the Apéry numbers, which Gessel showed satisfy a Lucas congruence in 1982.

Link to video: https://vimeo.com/644157702

Presented Via Zoom: https://rutgers.zoom.us/j/94346444480

Password: 6564120420

For further information see: https://sites.math.rutgers.edu/~zeilberg/expmath/