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

TL;DR

This paper investigates the asymptotic behavior of integers represented by specific binary forms derived from complex number powers, providing explicit constants for these forms.

Contribution

It computes the constants in the asymptotic formula for the number of integers represented by binary forms associated with powers of complex numbers.

Findings

Explicit formulas for constants $C_{R_n}$ and $C_{J_n}$.

Asymptotic count of integers represented by these forms.

Extension of previous results to new binary forms.

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 $R_F(Z) \sim C_FZ^{2/d}$ 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 + yi)^n = R_n(x, y) + J_n(x, y)i$, where the variables $x$ and $y$ are real.