Representation of integers by cyclotomic binary forms

TL;DR

This paper studies the integers represented by cyclotomic binary forms, proving finiteness of representations, describing their asymptotic distribution, and showing that the set of representable integers has density zero.

Contribution

It establishes bounds on the number and size of representations of integers by cyclotomic binary forms and analyzes their asymptotic distribution and density.

Findings

Number of representations of an integer is finite.

The set of integers represented has natural density zero.

Average number of representations grows like the square root of the logarithm of the integer.

Abstract

The homogeneous form $\Phi_n(X,Y)$ of degree $\varphi(n)$ which is associated with the cyclotomic polynomial $\phi_n(X)$ is dubbed a {\it cyclotomic binary form}. A positive integer $m\ge 1$ is said to be {\it representable by a cyclotomic binary form} if there exist integers $n,x,y$ with $n\ge 3$ and $\max\{|x|, |y|\}\ge 2$ such that $\Phi_n(x,y)=m$. We prove that the number $a_m$ of such representations of $m$ by a cyclotomic binary form is finite. More precisely, we have $\,\varphi(n) \le ({2}/ {\log 3})\log m\, $ and $\, \max\{|x|,|y|\} \le ({2}/{\sqrt{3}})\, m^{1/\varphi(n)}.\,$ We give a description of the asymptotic cardinality of the set of values taken by the forms for $n\geq 3$. This will imply that the set of integers $m$ such that $a_m\neq 0$ has natural density 0. We will deduce that the average value of the integers $a_m$ among the nonzero values of $a_m$ grows like $\sqrt{\log \, m}$.