On the Representation of Integers by Binary Forms Defined by Means of the Relation $(x + yi)^n = R_n(x, y) + J_n(x, y)i$

A. Mosunov

doi:10.2140/moscow.2022.11.71

paper

On the Representation of Integers by Binary Forms Defined by Means of the Relation $(x + y i)^{n} = R_{n} (x, y) + J_{n} (x, y) i$

arXiv:2110.04944 · doi:10.2140/moscow.2022.11.71

Abstract

Let $F$ be a binary form with integer coefficients, non-zero discriminant and degree $d \geq 3$ . 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.

On the Representation of Integers by Binary Forms Defined by Means of the Relation (x+yi)n=Rn​(x,y)+Jn​(x,y)i

Abstract

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On the Representation of Integers by Binary Forms Defined by Means of the Relation $(x + y i)^{n} = R_{n} (x, y) + J_{n} (x, y) i$