Lucas' theorem modulo $p^2$

TL;DR

This paper extends Lucas' theorem to modulo p^2, providing a characterization of digit pairs for which the theorem holds, using Pascal's triangle symmetries.

Contribution

It offers a new characterization of when Lucas' theorem applies modulo p^2, enhancing understanding of binomial coefficients in modular arithmetic.

Findings

Characterization of digit pairs for Lucas' theorem modulo p^2

Use of Pascal's triangle symmetries in the characterization

Extension of Lucas' theorem beyond modulo p

Abstract

Lucas' theorem describes how to reduce a binomial coefficient $\binom{a}{b}$ modulo $p$ by breaking off the least significant digits of $a$ and $b$ in base $p$. We characterize the pairs of these digits for which Lucas' theorem holds modulo $p^2$. This characterization is naturally expressed using symmetries of Pascal's triangle.