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On Quotients of a More General Theorem of Wilson

Ivan V. Morozov

TL;DR

This work generalizes Wilson's theorem by introducing a family of quotients $M_k(n)$ and their unsigned variants, and studies their arithmetic by examining sums $Z(n)$ and $Z^+(n)$. It derives closed forms for these sums, establishes Lehmer-type modular congruences involving Bernoulli and harmonic numbers, and develops generating functions that connect to Lerch transcendent and incomplete gamma functions. The results yield explicit parity-based formulas, prime-modulus congruences, and analytic representations, enriching the landscape of Wilson quotients and their interrelations. The inclusion of tables and generating functions supports potential primality insights and deeper structural understanding of these generalized 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.

On Quotients of a More General Theorem of Wilson

TL;DR

This work generalizes Wilson's theorem by introducing a family of quotients and their unsigned variants, and studies their arithmetic by examining sums and . It derives closed forms for these sums, establishes Lehmer-type modular congruences involving Bernoulli and harmonic numbers, and develops generating functions that connect to Lerch transcendent and incomplete gamma functions. The results yield explicit parity-based formulas, prime-modulus congruences, and analytic representations, enriching the landscape of Wilson quotients and their interrelations. The inclusion of tables and generating functions supports potential primality insights and deeper structural understanding of these generalized 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.

Paper Structure

This paper contains 11 sections, 3 theorems, 57 equations.

Table of Contents

  1. Introduction
  2. Sums of $M_{k}(n)$ and $M_{k}^{+}(n)$
  3. Formula for $Z(n)$
  4. Formula for $Z^{+}(n)$
  5. Modular Congruences
  6. Congruences of $M_{k}(p)$ and $M^{+}_{k}(p)$
  7. Congruences of $Z(n)$ and $Z^{+}(n)$
  8. Generating Functions
  9. Of $M$--numbers
  10. Of $Z$--numbers
  11. Tables

Key Result

Corollary 1

Given non--negative integers $n\geq 1$ and $k<n$, if and only if $n$ is non--composite.

Theorems & Definitions (5)

  • Corollary 1
  • Theorem 1
  • proof
  • Theorem 2
  • proof