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How often is $x\mapsto x^3$ one-to-one in $\mathbb{Z}/n\mathbb{Z}$?

Olivier Garet

TL;DR

This paper addresses the question of when the cubic map $x\mapsto x^3$ is a bijection on $\mathbb{Z}/n\mathbb{Z}$ and characterizes the corresponding set $W$. The authors show that $W$ consists precisely of square-free integers with no prime factor $p\equiv 1 \pmod{3}$ (i.e., all prime factors satisfy $p\equiv 2$ mod 3), and they prove the asymptotic $|W\cap\{1,\dots,n\}|\sim C\frac{n}{\sqrt{\log n}}$ with an explicit constant $C$. The method uses multiplicativity, Dirichlet series, and Delange's Tauberian theorem to convert local congruence restrictions into global density, with extensions to families defined by congruence constraints on prime factors. The result provides a concrete, provable parallel to Landau-type density phenomena and yields an exact value for the leading constant $C$.

Abstract

We characterize the integers n such that $x\mapsto x^3$ describes a bijection from the set $\mathbb{Z}/n\mathbb{Z}$ to itself and we determine the frequency of these integers. Precisely, denoting by $W$ the set of these integers, we prove that an integer belongs to $W$ if and only if it is square-free with no prime factor that is congruent to 1 modulo 3, and that there exists $C>0$ such that $$|W\cap\{1,\dots,n\}|\sim C\frac{n}{\sqrt{\log n}}\ .$$ These facts (or equivalent facts) are stated without proof on the OEIS website. We give the explicit value of $C$, which did not seem to be known. Analogous results are also proved for families of integers for which congruence classes for prime factors are imposed. The proofs are based on a Tauberian Theorem by Delange.

How often is $x\mapsto x^3$ one-to-one in $\mathbb{Z}/n\mathbb{Z}$?

TL;DR

This paper addresses the question of when the cubic map is a bijection on and characterizes the corresponding set . The authors show that consists precisely of square-free integers with no prime factor (i.e., all prime factors satisfy mod 3), and they prove the asymptotic with an explicit constant . The method uses multiplicativity, Dirichlet series, and Delange's Tauberian theorem to convert local congruence restrictions into global density, with extensions to families defined by congruence constraints on prime factors. The result provides a concrete, provable parallel to Landau-type density phenomena and yields an exact value for the leading constant .

Abstract

We characterize the integers n such that describes a bijection from the set to itself and we determine the frequency of these integers. Precisely, denoting by the set of these integers, we prove that an integer belongs to if and only if it is square-free with no prime factor that is congruent to 1 modulo 3, and that there exists such that These facts (or equivalent facts) are stated without proof on the OEIS website. We give the explicit value of , which did not seem to be known. Analogous results are also proved for families of integers for which congruence classes for prime factors are imposed. The proofs are based on a Tauberian Theorem by Delange.

Paper Structure

This paper contains 4 sections, 3 theorems, 12 equations.

Table of Contents

  1. Introduction
  2. Determination of W
  3. The elements of W are square-free
  4. Śtudy of prime factors

Key Result

Theorem 1

Let $Q$ be the set of square-free integers. For $m$ a positive natural integer and $A\subset \{1,\dots,m\}$, let $I_m(A)$ be the set of positive natural integers for which none of the prime factors is congruent modulo $m$ to an element of $A$. There exist non-zero positive constants $(c_a)_{1\le a\l and In the degenerate case where $\ell=\phi(m)$, we denote by $B$ the set of prime divisors of $m$

Theorems & Definitions (3)

  • Theorem 1
  • Theorem 2
  • Theorem 3