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On totally reducible binary forms: I

Authors:

C Hooley

Institution:

(1) School of Mathematics, Cardiff University, Senghennydd Road, PO Box 926, CF24 4YH Cardiff, UK

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.

Keywords:

Keyword" target="_blank">Keyword Binary forms

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