On totally reducible binary forms: I

Abstract

Letv(n) be the number of positive numbers up to a large limit n that are expressible in essentially more than one way by a binary formf that is a product ofl > 2 distinct linear factors with integral coefficients. We prove that

$$v(n) = O(n^{2/\ell - \eta _\ell + \in } )$$

, where

$$\eta \ell = \left\{ \begin{gathered} 1/\ell ^2 , if \ell = 3, \hfill \\ (\ell - 2)/\ell ^2 (\ell - 1), if \ell > 3 \hfill \\ \end{gathered} \right.$$

, thus demonstrating in particular that it is exceptional for a number represented byf to have essentially more than one representation.