Variations of Lucas' Theorem Modulo Prime Powers

TL;DR

This paper provides a simple inductive proof of a Lucas' theorem variation modulo prime powers, extending the result to small primes and offering a new characterization of Wolstenholme primes.

Contribution

The authors present an elementary inductive proof of a Lucas' theorem variation, extending it to primes 2 and 3 with modified exponents, and characterize Wolstenholme primes using Lucas-type congruences.

Findings

Proof based on Jacobsthal's congruence and binomial identities

Extension of the theorem to primes 2 and 3 with modified parameters

New characterization of Wolstenholme primes

Abstract

Let $p$ be a prime, and let $k,n,m,n_0$ and $m_0$ be nonnegative integers such that $k\ge 1$, and $_0$ and $m_0$ are both less than $p$. K. Davis and W. Webb established that for a prime $p\ge 5$ the following variation of Lucas' Theorem modulo prime powers holds $$ {np^k +n_0 \choose mp^k+m_0}\equiv{np^{\lfloor(k-1)/3\rfloor} \choose mp^{\lfloor(k-1)/3\rfloor}} {n_0 \choose m_0} \pmod{p^k}. $$ In the proof the authors used their earlier result that present a generalized version of Lucas' Theorem. In this paper we present a a simple inductive proof of the above congruence. Our proof is based on a classical congruence due to Jacobsthal, and we additionally use only some well known identities for binomial coefficients. Moreover, we prove that the assertion is also true for $p=2$ and $p=3$ if in the above congruence one replace $\lfloor(k-1)/3\rfloor$ by $\lfloor k/2\rfloor$, and by $\lfloor (k-1)/2\rfloor$, respectively. As an application, in terms of Lucas' type congruences, we obtain a new characterization of Wolstenholme primes.