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PreprintCC BY 4.0arXiv

Groups of Prime Units

  • Trey Li
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Abstract

We study groups of units modulo n whose nonidentity elements are all primes in {1,…,n−1}. We prove that there are only nine groups of prime units which are the full group of units modulo n, and there are fewer than 1022 groups of prime units containing all prime units in the possible range (√n,n−√n)∪{n−1}. We prove under Dickson's conjecture that there are infinitely many groups of prime units of arbitrary order d ≥ 2 modulo composite integers. We prove under Schinzel's Hypothesis H that there are infinitely many groups of prime units of order 3 modulo a prime r. We prove that the number of such prime moduli r smaller than X is ≪ X/(log X)3.

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Cite

BibTeXRIS
BibTeX
@misc{li2026groups,
  title = {{Groups of Prime Units}},
  author = {Trey Li},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.01793},
  eprint = {2611.01793},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.01793},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Groups of Prime Units
AU  - Trey Li
PY  - 2026
DA  - 2026/11/01
PB  - arXiv
DO  - 10.48550/arxiv.2611.01793
UR  - https://arxiv.org/abs/2611.01793
AB  - We study groups of units modulo n whose nonidentity elements are all primes in {1,…,n−1}. We prove that there are only nine groups of prime units which are the full group of units modulo n, and there are fewer than 10^22 groups of prime units containing all prime units in the possible range (√n,n−√n)∪{n−1}. We prove under Dickson's conjecture that there are infinitely many groups of prime units of arbitrary order d ≥ 2 modulo composite integers. We prove under Schinzel's Hypothesis H that there are infinitely many groups of prime units of order 3 modulo a prime r. We prove that the number of such prime moduli r smaller than X is ≪ X/(log X)^3.
ER  -