On certain Diophantine equations of the form z2 = f(x)2 ± g(y)2

Abstract

In this note we study the existence of integer solutions of the Diophantine equationz2=f(x)2±g(y)2 for certain polynomials f,g∈Z[x] of degree ≥3. In particular, for given k∈N we prove that for all a,b∈Z satisfying the condition a2+b2≠0, the above Diophantine equation, with f(x)=xk(x+a), g(x)=xk(x+b) and any choice of the sign, has infinitely many solutions in integers x,y,z such that f(x)g(y)≠0. Moreover, we prove that for f(x)=x3 and g(x)=x(x+1)(x+2) the system of Diophantine equationsz12=f(x)2+g(y)2,z22=f(x)2+g(y+1)2 has infinitely many solutions in positive integers x,y,z1,z2 with gcd⁡(x,y)=1. Similar result is proved for the systemz12=f(x)2+g(y)2,z22=f(x+1)2+g(y)2. We also present some experimental results concerning the construction of polynomials with rational coefficients for which the Diophantine equation z2=f(x)2±g(y)2 has infinitely many solutions in integers.