Generalized Lucas Theorem

Abstract

Let $p$ be a prime. Let $A$ and $B$, $A \ge B \ge 0$, be integers with base $p$ expansions $A = \alpha_i\alpha_{i-1}\dots \alpha_0$ and $B = \beta_i\beta_{i-1}\dots \beta_0$. Lucas proved that $$\binom{A}{B} \equiv \prod_{j=0}^{j=i}\binom{\alpha_j}{\beta_j} \text{ mod } p.$$ Similarly as proved by Kummer, the $p$-adic valuation $v_p\binom{A}{B}$ is the number of borrows when computing $A-B$ in base $p$, or the number of carries in $(A-B)+B$ in base $p$. Davis and Webb discovered a generalization of Lucas's Theorem for prime powers. We prove a similar generalization in a different form using the concept of pseudo-digits.