A diophantine sum with factorials

TL;DR

This paper explores solutions to a Diophantine equation involving factorials, expressing it as a cubic form or sum of binomial coefficients, and relates some solutions to Fermat's Last Theorem.

Contribution

It introduces new solutions to factorial-based Diophantine equations and connects them to higher degree forms and Fermat's Last Theorem.

Findings

Solutions to factorial Diophantine equations as cubic forms

Representation of solutions as sums of binomial coefficients

Connections between solutions and Fermat's Last Theorem

Abstract

We give solutions of a Diophantine equation containing factorials, which can be written as a cubic form, or as a sum of binomial coefficients. We also give some solutions to higher degree forms and relate some solutions to an unsolvable Fermat Last Theorem equation.