Generalized Lucas Theorem

TL;DR

This paper generalizes Lucas's Theorem for binomial coefficients modulo prime powers using the concept of pseudo-digits, extending classical results and providing new insights into p-adic valuations.

Contribution

It introduces a novel generalization of Lucas's Theorem for prime powers employing pseudo-digits, expanding the theoretical framework of binomial coefficient congruences.

Findings

Established a new form of Lucas's Theorem for prime powers.

Connected p-adic valuation with borrow and carry operations in base p.

Extended classical combinatorial congruences to broader prime power contexts.

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.