On a Conjecture of Sun Involving Powers of Three

Abstract

Given a positive integer n ≥ 2, let D(n) denote the smallest positive integer m such that a³ + a (a = 1, ⋯, n) are pairwise distinct modulo m². A conjecture of Sun states that D(n) = 3^k, where 3^k is the least power of 3 no less than

\sqrt{n}

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\sqrt n}$$\end{document}

. The purpose of this paper is to confirm this conjecture.

Keywords

Congruence / Gauss sum / Quadratic residue / Kloosterman sum / 11A07 / 11L05

References

[1] Arnold L K, Benkoski S J, McCabe B J. The discriminator (a simple application of Bertrand’s postulate). Amer. Math. Monthly, 1985, 92: 275-277

[2] Bremser P S, Schumer P D, Washinyton L C. A note on the incongruence of consecutive integers to a fixed power. J. Number Theory, 1990, 35: 105-108

[3] Hua L K. Introduction to Number Theory, 1982, Berlin, Heidelberg, New York, Springer-Verlag

[4] Ireland K, Rosen M. A Classical Introduction to Mordern Number Theory, 19903rd ed.New York, Springer-Verlag 84

[5] Iwaniec H. Topic in Classical Automorphic Forms, 1997, Providence, RI, Amer. Math. Soc.

[6] Moree P. The incongruence of consecutive values of polynomials. Finite Fields Appl., 1996, 2: 321-335

[7] Moree P, Mullen G L. Dickson polynomial discriminators. J. Number Theory, 1996, 59: 88-105

[8] Sun Z-W. On functions taking only prime values. J. Number Theory, 2013, 133: 2794-2812

[9] Sun Z-W. New Conjectures in Number Theory and Combinatorics, 2021, Harbin, Harbin Institute of Technology Press

[10] Zieve M. A note on discriminator. J. Number Theory, 1998, 73: 122-138

RIGHTS & PERMISSIONS

The Editorial Office of CAM and Springer-Verlag Berlin Heidelberg

Just Accepted

This article has successfully passed peer review and final editorial review, and will soon enter typesetting, proofreading and other publishing processes. The currently displayed version is the accepted final manuscript. The officially published version will be updated with format, DOI and citation information upon launch. We recommend that you pay attention to subsequent journal notifications and preferentially cite the officially published version. Thank you for your support and cooperation.