Diophantus Equations and Partially Ordered Sets

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 $A_{n+2}=aA_{n+1}+bA_{n}$.