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PreprintCC BY 4.0arXiv

Results for distinct covering systems with a prime minimum modulus q ≥ 11

  • Michael Filaseta
  • Alexandros Kalogirou
  • Jonah Klein
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Abstract

We prove that if the smallest modulus of a covering system is a prime q ≥ 11, then the sum of the reciprocals of the moduli is at least 1+10−28. This improves on a previous more general bound of the first two authors in this particular case. We also show that it is impossible to construct a covering system using distinct multiples of q as moduli. Finally, we show as a corollary that it is impossible to construct $10$ distinct covering systems with the property that no modulus is used in more than one of the covering systems.

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Cite

BibTeXRIS
BibTeX
@misc{filaseta2026results,
  title = {{Results for distinct covering systems with a prime minimum modulus $q \ge 11$}},
  author = {Michael Filaseta and Alexandros Kalogirou and Jonah Klein},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.34541},
  eprint = {2610.34541},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.34541},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Results for distinct covering systems with a prime minimum modulus q ≥ 11
AU  - Michael Filaseta
AU  - Alexandros Kalogirou
AU  - Jonah Klein
PY  - 2026
DA  - 2026/10/25
PB  - arXiv
DO  - 10.48550/arxiv.2610.34541
UR  - https://arxiv.org/abs/2610.34541
AB  - We prove that if the smallest modulus of a covering system is a prime q ≥ 11, then the sum of the reciprocals of the moduli is at least 1+10^−28. This improves on a previous more general bound of the first two authors in this particular case. We also show that it is impossible to construct a covering system using distinct multiples of q as moduli. Finally, we show as a corollary that it is impossible to construct $10$ distinct covering systems with the property that no modulus is used in more than one of the covering systems.
ER  -