Representation of integers by cyclotomic binary forms
Volume 184 / 2018
Abstract
The homogeneous form $\varPhi_n(X,Y)$ of degree $\varphi(n)$ which is associated with the cyclotomic polynomial $\phi_n(X)$ is dubbed a cyclotomic binary form. A positive integer $m\ge 1$ is said to be 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 $\varPhi_n(x,y)=m$. 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 nonzero values of $a_m$ grows like $\sqrt{\log \, m}$.
