Diophantus Equations and Partially Ordered Sets

TL;DR

This paper solves new classes of factorial-based Diophantine equations, establishes conditions for monotonicity in partially ordered sets, and extends previous results to Fibonacci and linear recurrence sequences.

Contribution

It finds all solutions to complex factorial Diophantine equations, introduces a sufficient condition for monotonicity in posets, and generalizes earlier work to Fibonacci and linear recurrence sequences.

Findings

Solved factorial Diophantine equations with multiple variables.

Established a condition for monotonic functions on posets.

Extended results to Fibonacci and linear recurrence sequences.

Abstract

In [1] it is shown that the Diophantine equation $(k!)^n+k^n=(n!)^k+n^k$ only has the trivial solution $n=k$, and $(k!)^n-k^n=(n!)^k-n^k$ only has the solutions $n=k$, $(n, k)=(1, 2),$ and $(2, 1)$. In this article we find all solutions of the Diophantine Equations $a_1!a_2!\cdots a_n! \pm a_1a_2 \cdots a_n = b_1!b_2! \cdots b_k! \pm b_1b_2 \cdots b_k$, where $a_i$ majorizes $b_i$. Furthermore we find a sufficient condition on a function $f:N\to R^+$ to guarantee that $f$ gives a monotone function on the POSET of all finite sequences of natural numbers. We then use that to solve other Diophantine equations involving factorials and generalize the results of [2]. We also explore similar Diophantine Equations for the Fibonacci Sequence and other sequences of natural numbers given by linear recursions of the form $A_{n+2}=aA_{n+1}+bA_{n}$.