Conjectures about Primes and Cyclic Numbers
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915441270587392 |
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| author | Cohen, Joel E. |
| author_facet | Cohen, Joel E. |
| contents | A positive integer $n$ is defined to be cyclic if and only if every group of size $n$ is cyclic. Equivalently, $n$ is cyclic if and only if $n$ is relatively prime to the number of positive integers less than $n$ that are relatively prime to $n$. Because every prime number is cyclic, it is natural to ask whether a (proved or conjectured) property of primes extends to cyclic numbers. I review proved or conjectured properties of primes (including some new conjectures about primes) and propose analogous conjectures about cyclic numbers. Using the 28,488,167 cyclic numbers less than $10^8$, I test the conjectures about cyclic numbers and disprove the cyclic analog of the second conjecture about primes of Hardy and Littlewood. Proofs or disproofs of the remaining conjectures are invited. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2508_08335 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Conjectures about Primes and Cyclic Numbers Cohen, Joel E. Number Theory 11Y99 (Primary) 11Y11, 11Y55 (Secondary) A positive integer $n$ is defined to be cyclic if and only if every group of size $n$ is cyclic. Equivalently, $n$ is cyclic if and only if $n$ is relatively prime to the number of positive integers less than $n$ that are relatively prime to $n$. Because every prime number is cyclic, it is natural to ask whether a (proved or conjectured) property of primes extends to cyclic numbers. I review proved or conjectured properties of primes (including some new conjectures about primes) and propose analogous conjectures about cyclic numbers. Using the 28,488,167 cyclic numbers less than $10^8$, I test the conjectures about cyclic numbers and disprove the cyclic analog of the second conjecture about primes of Hardy and Littlewood. Proofs or disproofs of the remaining conjectures are invited. |
| title | Conjectures about Primes and Cyclic Numbers |
| topic | Number Theory 11Y99 (Primary) 11Y11, 11Y55 (Secondary) |
| url | https://arxiv.org/abs/2508.08335 |