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    • On totally reducible binary forms: I

      C Hooley

      Volume 111 Issue 3 August 2001 pp 249-262

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      Permanent link:
      https://www.ias.ac.in/article/fulltext/pmsc/111/03/0249-0262

    • Keywords

       

      Binary forms

    • 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.

    • Author Affiliations

       

      C Hooley1

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

    • Dates

       
      Published
      August 2001
      Manuscript received
      18 October 2000
      Manuscript revised
      7 February 2001

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