• On totally reducible binary forms: I

• Fulltext

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 &gt; 2 distinct linear factors with integral coefficients. We prove that$$v(n) = O(n^{2/\ell - \eta _\ell + \in } )$$, where$$\eta \ell = \left\{ \begin{gathered} 1/\ell ^2 , if \ell = 3, \hfill \\ (\ell - 2)/\ell ^2 (\ell - 1), if \ell &gt; 3 \hfill \\ \end{gathered} \right.$$, thus demonstrating in particular that it is exceptional for a number represented byf to have essentially more than one representation.

• Author Affiliations

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

• Proceedings – Mathematical Sciences

Volume 132, 2022
All articles
Continuous Article Publishing mode

• Editorial Note on Continuous Article Publication

Posted on July 25, 2019