Vol. 11, No. 1, 2022

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On the representation of integers by binary forms defined by means of the relation $(x + yi)^n = R_n(x, y) + J_n(x, y)i$

Anton Mosunov

Vol. 11 (2022), No. 1, 71–78
Abstract

Let F be a binary form with integer coefficients, degree d 3 and nonzero discriminant. Let RF(Z) denote the number of integers of absolute value at most Z which are represented by F. In 2019 Stewart and Xiao proved that RF(Z) CFZ2d for some positive number CF. We compute CRn and CJn for the binary forms Rn(x,y) and Jn(x,y) defined by means of the relation

(x + yi)n = R n(x,y) + Jn(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