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 $a_1!a_2!\cdots a_n!\pm a_1{a}_2\cdots a_n=b_1!b_2!\cdots b_k!\pm b_1{b}_2\cdots b_k$, 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 $A_{n+2}=a{A}_{n+1}+b{A}_n$.

arXiv comment12 pages, 1 figure

References in corpus (1)

• Explicit Bounds For The Diophantine Equation A!B! = C!