The Diophantine equation 𝑓(𝑥)=𝑔(𝑦)

Abstract

Let f ( x ) , g ( y ) f(x),g(y) be polynomials over Z \mathbb {Z} of degrees n n and m m respectively and with leading coefficients a n , b m {a_n},{b_m} . Suppose that m | n m|n and that a n / b m {a_n}/{b_m} is the m m th power of a rational number. We give two elementary proofs that the equation f ( x ) = g ( y ) f(x) = g(y) has at most finitely many integral solutions unless f ( x ) = g ( h ( x ) ) f(x) = g(h(x)) for some polynomial h ( x ) h(x) with rational coefficients taking integral values at infinitely many integers.