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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2504.13511 |
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| _version_ | 1866915249925390336 |
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| author | Garet, Olivier |
| author_facet | Garet, Olivier |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_13511 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | How often is $x\mapsto x^3$ one-to-one in $\mathbb{Z}/n\mathbb{Z}$? Garet, Olivier Number Theory 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. |
| title | How often is $x\mapsto x^3$ one-to-one in $\mathbb{Z}/n\mathbb{Z}$? |
| topic | Number Theory |
| url | https://arxiv.org/abs/2504.13511 |