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

In this paper we investigate Diophantine equations of the form $T^2=G(\bar{X}),\bar{X}=(X_1,\dots ,X_m)$, where $m=3$ or $m=4$ and $G$ is specific homogenous quintic form. First, we prove that if $F(x,y,z)=x^2+y^2+a{z}^2+bxy+cyz+dxz\in \mathbb{Z}[x,y,z]$ and $(b-2,4a-d^2,d)\neq (0,0,0)$,thentheDiophantineequation$t^2=nxyzF(x,y,z)$ 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 $n\in \text{Q}\setminus \{0\}$ 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 $T^2=a{X}_1^5+b{X}_2^5+c{X}_3^5+d{X}_4^5$, where $a, b, c, d\in\Z$\mathrm{.}Inparticular,weprovethatforeach$m, n\in\Z\setminus{0},$ 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 $\mathbb{Z}[t]$. Moreover, we show how modification of the presented method can be used in order to prove that for each $n\in \text{Q}\setminus \{0\}$, 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 $\mathbb{Z}[t]$.