Seminars & Colloquia Calendar

Experimental Mathematics Seminar

Lucas congruences modulo p2

Eric Rowland, Hofstra University

Location:  Zoom
Date & time: Thursday, 04 November 2021 at 5:00PM - 6:00PM

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 ApÃ©ry numbers, which Gessel showed satisfy a Lucas congruence in 1982