On primitive integer solutions of the Diophantine equation t2=G(x,y,z)t2=G(x,y,z) and related results

In this paper we investigate Diophantine equations of the form T2=G(X‾), X‾=(X1,…,Xm), where m=3m=3 or m=4m=4 and G   is a specific homogeneous quintic form. First, we prove that if F(x,y,z)=x2+y2+az2+bxy+cyz+dxz∈Z[x,y,z]F(x,y,z)=x2+y2+az2+bxy+cyz+dxz∈Z[x,y,z] and (b−2,4a−d2,d)≠(0,0,0)(b−2,4a−d2,d)≠(0,0,0), then for all n∈Z∖{0}n∈Z∖{0} the Diophantine equation t2=nxyzF(x,y,z)t2=nxyzF(x,y,z) has a solution in polynomials x, y, z, t   with integer coefficients, with no polynomial common factor of positive degree. In case a=d=0a=d=0, b=2b=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∈Q∖{0}n∈Q∖{0} the Diophantine equationT2=n(X15+X25+X35+X45) has a solution in co-prime (non-homogeneous) polynomials in two variables with integer coefficients. We also present a method which sometimes allows us to prove the existence of primitive integer solutions of more general quintic Diophantine equation of the form T2=aX15+bX25+cX35+dX45, where a,b,c,d∈Za,b,c,d∈Z. In particular, we prove that for each m,n∈Z∖{0}m,n∈Z∖{0}, the Diophantine equationT2=m(X15−X25)+n2(X35−X45) has a solution in polynomials which are co-prime over Z[t]Z[t]. Moreover, we show how a modification of the presented method can be used in order to prove that for each n∈Q∖{0}n∈Q∖{0}, the Diophantine equationt2=n(X15+X25−2X35) has a solution in polynomials which are co-prime over Z[t]Z[t].