The residue-counts of $x^2+a/x$ modulo a prime

TL;DR

This paper derives an explicit formula for the residue counts of the function $x^2 + a/x$ modulo a prime, connecting it with cubic residues and quadratic forms, advancing understanding of these algebraic structures.

Contribution

It provides the first explicit formula for the residue counts of $x^2 + a/x$ modulo a prime, linking it to cubic residues and quadratic forms.

Findings

Explicit formula for $V_p(x^2 + a/x)$ derived

Connection established between residue counts and cubic residues

Results enhance understanding of algebraic structures modulo primes

Abstract

For a prime $p>3$ and $a\in \Bbb Z$ with $p\nmid a$ let $V_p(x^2+\frac ax)$ be the residue-counts of $x^2+\frac ax$ modulo $p$ as $x$ runs over $1,2,\ldots,p-1$. In this paper, we obtain an explicit formula for $V_p(x^2+\frac ax)$, which is concerned with cubic residues and binary quadratic forms.