Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Representation of integers by cyclotomic binary forms   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 $\sqrt{\log \, m}$.

24
The categories of this paper are: dg-ga
5
The categories of this paper are: gr-qc
23
The categories of this paper are: solv-int
27
The categories of this paper are: comp-gas
12
The categories of this paper are: nucl-th
28
The categories of this paper are: alg-geom
25
The categories of this paper are: patt-sol
0
The categories of this paper are: hep-ph
29
The categories of this paper are: funct-an
4
The categories of this paper are: cond-mat
7
The categories of this paper are: hep-th
8
The categories of this paper are: hep-ex
6
The categories of this paper are: astro-ph
10
The categories of this paper are: q-bio
15
The categories of this paper are: nucl-ex
35
The categories of this paper are: mtrl-th