LAUSR

Let $$f\in \mathbb {Q}[x]$$ f ∈ Q [ x ] be a polynomial without multiple roots and $$\deg {f}\ge 2$$ deg f ≥ 2 . We give conditions for $$f=x^2+bx+c$$ f = x 2 + b x + c under which the Diophantine equation $$2f(x)f(y)=f(z)(f(x)+f(y))$$ 2 f ( x ) f ( y ) = f ( z ) ( f ( x ) + f ( y ) ) has infinitely many nontrivial integer solutions and prove that this equation has infinitely many rational parametric solutions for $$f=x^2+bx$$ f = x 2 + b x with nonzero integer b . Moreover, we show that it has a rational parametric solution for infinitely many cubic polynomials.