Generalized Lucas Theorem

TL;DR

The paper tackles the problem of extending Lucas's binomial congruence to prime powers by introducing pseudo-digits, a block-structured decomposition of $A$ and $B$ in base $p$. It develops a main theorem that expresses $\binom{A}{B}$ modulo $p^{n+m}$ (where $m = v_p\binom{A}{B}$) in terms of products of binomial coefficients formed from blocks of $n$ pseudo-digits and adjacent shorter blocks, with a rigorous inductive proof across the cases determined by the least significant pseudo-digits. A key lemma connects the $p$-adic valuations of these block binomials to the sum of valuations of individual digits, enabling the induction to go through and the congruence to hold. The work also compares the pseudo-digit formulation with the Davis–Webb generalization for prime powers, showing their equivalence and providing practical guidance through worked examples. Overall, the paper broadens Lucas’s theorem to higher powers of $p$ and offers a structured, computable framework for binomial coefficients modulo $p^{n+m}$ with clear connections to p-adic carry/borrow concepts.

Abstract

Let $p$ be a prime. Let $A$ and $B$, $A \ge B \ge 0$, be integers with base $p$ expansions $A = α_iα_{i-1}\dots α_0$ and $B = β_iβ_{i-1}\dots β_0$. Lucas proved that $$\binom{A}{B} \equiv \prod_{j=0}^{j=i}\binom{α_j}{β_j} \text{ mod } p.$$ 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.

Theorems & Definitions (2)

• Theorem 1.1
• Lemma 2.1