A note on the Diophantine equation $$f(x)f(y)=f(z^2)$$ f ( x ) f ( y ) = f ( z 2 )

Abstract

Let \(f\in {\mathbb {Q}}[X]\) be a polynomial without multiple roots and \(deg(f)\ge 2\). We consider the Diophantine equation \(f(x)f(y)=f(z^2)\). For two classes of irreducible quadratic polynomials, this equation has infinitely many nontrivial integer solutions, if the corresponding Pell’s equations satisfy a condition. For a special cubic polynomial, it has a one-parameter family of rational solutions. For \(f(X)=X(X^2+X+k)\) and \(X(X^2+kX+1)\) there are infinitely many \(k \in {\mathbb {Q}}\) such that the title equation has rational solutions.