# A polynomial variant of a problem of Diophantus and its consequences

## TL;DR

This paper proves that polynomial Diophantine quadruples over real polynomials are necessarily regular, leading to new non-existence results for polynomial Diophantine sets over integers with certain properties.

## Contribution

It establishes a polynomial analogue of a classical Diophantine problem, showing all quadruples are regular and deriving non-existence results over integers.

## Key findings

- All polynomial Diophantine quadruples in 4[X]4 are regular.
- No quadruples over 4[X]4 with certain properties exist over 4[4]4.
- No five polynomials over 4[4]4 satisfy the polynomial Diophantine condition with positive integer n.

## Abstract

We prove that every Diophantine quadruple in $\mathbb{R}[X]$ is regular. More precisely, we prove that if $\{a, b, c, d\}$ is a set of four non-zero polynomials from $\mathbb{R}[X]$, not all constant, such that the product of any two of its distinct elements increased by $1$ is a square of a polynomial from $\mathbb{R}[X]$, then $$(a+b-c-d)^2=4(ab+1)(cd+1).$$ One consequence of this result is that there does not exist a set of four non-zero polynomials from $\mathbb{Z}[X]$, not all constant, such that a product of any two of them increased by a positive integer $n$, which is not a perfect square, is a square of a polynomial from $\mathbb{Z}[X]$. Our result also implies that there does not exist a set of five non-zero polynomials from $\mathbb{Z}[X]$, not all constant, such that a product of any two of them increased by a positive integer $n$, which is a perfect square, is a square of a polynomial from $\mathbb{Z}[X]$.