On Quotients of a More General Theorem of Wilson

TL;DR

This paper extends Wilson's theorem to generate a broader class of quotients, explores their relationships, and derives new properties including sums, congruences, and generating functions.

Contribution

It introduces a generalized corollary of Wilson's theorem that produces many new quotients and analyzes their properties and interrelations.

Findings

Derived expressions for sums of the quotients

Established modular congruences extending Lehmer's results

Developed generating functions for the quotients

Abstract

The basis of this work is a simple, extended corollary of Wilson's theorem. This corollary generates many more quotients than those already generated by Wilson's theorem, and it was of interest to derive how they relate to each other and build on the established properties of the original quotients. The most important results that were found were expressions for sums of these quotients, modular congruences that extended the results of Lehmer, and generating functions.