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 $\sqrt{ \log \, m}$.
Categories: math.NT