Representation of integers by cyclotomic binary forms

2017/12/25 by Fouvry, Etienne, Levesque, Claude, Waldschmidt, Michel · 1 citation
#11E76 12E10 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1712.09019

Abstract

The homogeneous form Φ_n(X,Y) of degree φ(n) which is associated with the cyclotomic polynomial ϕ_n(X) is dubbed a \it cyclotomic binary form. A positive integer m≥ 1 is said to be \it representable by a cyclotomic binary form if there exist integers n,x,y with n≥ 3 and max\|x|, |y|\≥ 2 such that Φ_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 φ(n) ≤ (2/ log 3)log m and max\|x|,|y|\ ≤ (2/√(3)) m^1/φ(n). We give a description of the asymptotic cardinality of the set of values taken by the forms for n≥ 3. This will imply that the set of integers m such that a_m≠ 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 √(log m).

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