Periodic minimum in the count of binomial coefficients not divisible by a prime

TL;DR

This paper investigates the periodic behavior of binomial coefficients not divisible by a prime, proposing a method to identify their minimum points and solving a longstanding conjecture in the field.

Contribution

It introduces a general approach to locate minima in the periodic functions of binomial coefficients modulo a prime, addressing a specific conjecture by Wilson.

Findings

Identified the minimum points in the periodic count of binomial coefficients not divisible by a prime.

Provided a solution to Wilson's conjecture on the asymptotic behavior of Pascal's triangle modulo a prime.

Enhanced understanding of the oscillatory nature of binomial coefficients in modular arithmetic.

Abstract

The summatory function of the number of binomial coefficients not divisible by a prime is known to exhibit regular periodic oscillations, yet identifying the less regularly behaved minimum of the underlying periodic functions has been open for almost all cases. We propose an approach to identify such minimum in some generality, solving particularly a previous conjecture of B. Wilson [Asymptotic behavior of Pascal's triangle modulo a prime, Acta Arith. 83 (1998), pp. 105-116].