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- W4301630733 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 $mge 1$ is said to be {it representable by a cyclotomic binary form} if there exist integers $n,x,y$ with $nge 3$ 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 $ngeq 3$. This will imply that the set of integers $m$ such that $a_mneq 0$ 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 $sqrt{log , m}$." @default.
- W4301630733 created "2022-10-05" @default.
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- W4301630733 date "2017-12-25" @default.
- W4301630733 modified "2023-09-30" @default.
- W4301630733 title "Representation of integers by cyclotomic binary forms" @default.
- W4301630733 doi "https://doi.org/10.48550/arxiv.1712.09019" @default.
- W4301630733 hasPublicationYear "2017" @default.
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