Banach matchboxes problem and a congruence for primes

TL;DR

This paper explores a probability problem involving matchboxes and derives a surprising prime congruence related to binomial coefficients and powers of two, revealing new number-theoretic insights.

Contribution

It introduces a novel congruence for odd primes based on an identity from the Banach matchboxes problem, linking probability and number theory.

Findings

Established a new prime congruence involving binomial coefficients and powers of two

Connected a classical probability problem to prime number properties

Provided a proof of the congruence using an identity from the matchboxes problem

Abstract

Using an identity arising in the known Banach probability problem on matchboxes, we prove an unexpected congruence for odd prime $p:$ for $1\leq k\leq \frac{p-1}{2},\enskip \sum_{i=1}^{p-2k-1}2^{i-1}\binom{k-1+i}{k}\equiv 0\pmod p.$