On the representation of integers by binary forms defined by means of the relation (x + yi)n = Rn(x,y) + Jn(x,y)i

Let $F$ be a binary form with integer coefficients, degree $d \geq 3$ and nonzero discriminant. Let $R_{F} (Z)$ denote the number of integers of absolute value at most $Z$ which are represented by $F$ . In 2019 Stewart and Xiao proved that $R_{F} (Z) \sim C_{F} Z^{2 ∕ d}$ for some positive number $C_{F}$ . We compute $C_{R_{n}}$ and $C_{J_{n}}$ for the binary forms $R_{n} (x, y)$ and $J_{n} (x, y)$ defined by means of the relation

{(x + y i)}^{n} = R_{n} (x, y) + J_{n} (x, y) i,

where the variables $x$ and $y$ are real.

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Keywords

binary form, automorphism group, representation of integers by binary forms, fundamental region, Thue equation

Mathematical Subject Classification

Primary: 11E76, 11D59, 11D45

Milestones

Received: 4 August 2021

Revised: 19 February 2022

Accepted: 5 March 2022

Published: 30 March 2022

Authors

Anton Mosunov
University of Waterloo Waterloo, ON Canada

Vol. 11, No. 1, 2022

Anton Mosunov

Abstract

PDF Access Denied

Keywords

Mathematical Subject Classification

Milestones

Authors