On primitive integer solutions of the Diophantine equation $t^2=G(x,y,z)$ and related results

TL;DR

This paper investigates solutions to specific quintic Diophantine equations, proving existence of polynomial solutions and providing methods to find primitive integer solutions for various forms, including equations involving sums and differences of fifth powers.

Contribution

The paper introduces new methods to establish the existence of primitive polynomial solutions for certain quintic Diophantine equations, extending previous results and providing explicit solution constructions.

Findings

Proved polynomial solutions exist for equations of the form t^2 = n xyz F(x,y,z).

Established infinite primitive solutions for specific equations like T^2 = n(X_1^5 + X_2^5 + X_3^5 + X_4^5).

Developed methods to find solutions for more general quintic equations involving linear combinations of fifth powers.

Abstract

In this paper we investigate Diophantine equations of the form $T^2=G(\overline{X}),\; \overline{X}=(X_{1},\ldots,X_{m})$, where $m=3$ or $m=4$ and $G$ is specific homogenous quintic form. First, we prove that if $F(x,y,z)=x^2+y^2+az^2+bxy+cyz+dxz\in\Z[x,y,z]$ and $(b-2,4a-d^2,d)\neq (0,0,0)$, then the Diophantine equation $t^2=nxyzF(x,y,z)$ has solution in polynomials $x, y, z, t$ with integer coefficients, without polynomial common factor of positive degree. In case $a=d=0, b=2$ we prove that there are infinitely many primitive integer solutions of the Diophantine equation under consideration. As an application of our result we prove that for each $n\in\Q\setminus\{0\}$ the Diophantine equation \begin{equation*} T^2=n(X_{1}^5+X_{2}^5+X_{3}^5+X_{4}^5) \end{equation*} has a solution in co-prime (non-homogenous) polynomials in two variables with integer coefficients. We also present a method which sometimes allow us to prove the existence of primitive integers solutions of more general quintic Diophantine equation of the form $T^2=aX_{1}^5+bX_{2}^5+cX_{3}^5+dX_{4}^5$, where $a, b, c, d\in\Z$. In particular, we prove that for each $m, n\in\Z\setminus\{0\},$ the Diophantine equation \begin{equation*} T^2=m(X_{1}^5-X_{2}^5)+n^2(X_{3}^5-X_{4}^5) \end{equation*} has a solution in polynomials which are co-prime over $\Z[t]$. Moreover, we show how modification of the presented method can be used in order to prove that for each $n\in\Q\setminus\{0\}$, the Diophantine equation \begin{equation*} t^2=n(X_{1}^5+X_{2}^5-2X_{3}^5) \end{equation*} has a solution in polynomials which are co-prime over $\Z[t]$.