Representation of integers by cyclotomic binary forms

Étienne Fouvry¹ ; Claude Levesque² ; Michel Waldschmidt³

¹ Institut de Mathématique d’Orsay Université Paris-Sud CNRS, Université Paris-Saclay F-91405 Orsay, France
² Département de mathématiques et de statistique Université Laval Québec, QC, Canada G1V 0A6
³ Sorbonne Université UPMC Univ Paris 06 UMR 7586 IMJ-PRG F-75005 Paris, France

Acta Arithmetica, Tome 184 (2018) no. 1, pp. 67-86.

Résumé

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 $\sqrt{\log \, m}$.

Détail

DOI : 10.4064/aa171012-24-12

Keywords: homogeneous form varphi degree varphi which associated cyclotomic polynomial phi dubbed cyclotomic binary form positive integer said representable cyclotomic binary form there exist integers max varphi prove number representations cyclotomic binary form finite precisely have varphi log log max sqrt varphi description asymptotic cardinality set values taken forms geq imply set integers neq has natural density deduce average value nonzero values nbsp grows sqrt log