On a Discriminator for the Polynomial $f(x)=x^3+x$

Abstract

Let $\Delta (n)$ denote the smallest positive integer m such that $a^3+a(1\leqslant a\leqslant n)$ are pairwise distinct modulo m. The purpose of this paper is to determine $\Delta (n)$ for all positive integers n.

Keywords

Congruence / Character sum / Quadratic residue / Kloosterman sum

References

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Funding

National Natural Science Foundation of China(11922113)