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

Zhi-Hong Sun

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

Number Theory 2023-09-15 v1

Authors: Zhi-Hong Sun

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.

Keywords

residue-counts prime modulus cubic residues binary quadratic forms explicit formula

Cite

@article{arxiv.2309.07685,
  title  = {The residue-counts of $x^2+a/x$ modulo a prime},
  author = {Zhi-Hong Sun},
  journal= {arXiv preprint arXiv:2309.07685},
  year   = {2023}
}

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8 pages

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