Polynomial squares of the form \(aX^m+b(1-X)^n+c\)
Summary: It is an open diophantine problem to prove the finiteness of the integral solutions for equations like \(y^2=3^m+2^n+1\). This may be shown to correspond to one of the simplest unknown cases of a conjecture by Lang and Vojta and is probably very difficult; actually also the most basic analogous questions over function fields seem not to follow easily from known principles. With this in mind, in the present note we consider the equation \(y(X)^2=aX^m + b(1-2X)^n+c\), to be solved in the unknowns \(a,b,c\in \mathbb{C}^*\), \(m,n\in\mathbb{N}\) and polynomials \(y\in\mathbb{C}[X]\); we show that only pairs \((m , n)\) with \(m,n\leq 8\) may give rise to solutions. Our arguments are somewhat ad hoc and in a way surprising; the problem is left to find a more natural and general approach.
scientific article; zbMATH DE number 5121476