Continuant Diophantine equations

TL;DR

This paper explores a class of Diophantine equations involving continuant functions, revealing their solution structures and methods to generate new solutions, with connections to polynomial factorizations.

Contribution

It introduces a new family of Diophantine equations involving continuant functions and analyzes their solution structures and generation methods.

Findings

Solution sets have a rich, structured nature.

Methods for generating new solutions from existing ones.

Connections between solutions and polynomial factorizations.

Abstract

We investigate a family of Diophantine polynomial equations which involve continuant functions. In particular, given a polynomial $P(x)\in \mathbb{Z}[x]$ and $n\in \mathbb{N}$, we consider the equation $P(K_n(x_1,\ldots, x_n)) = K_{n+1}(x_0,\ldots,x_n)K_{n+1}(x_1,\ldots, x_{n+1})$. We show that with certain restrictions on $P(x)$ the set of its solutions has a rich structure. In particular, we provide several ways of generating new solutions from the existing ones. In the last section we discuss the relation between the solutions of the above Diophantine equation for arbitrary values of $n$ and factorisations $P(m) = d_1d_2$ for integers $m,d_1$ and $d_2$.