Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2610.34541

Results for distinct covering systems with a prime minimum modulus $q \ge 11$

Michael Filaseta·Alexandros Kalogirou·Jonah Klein
View original source ↗Read source document ↗Export metadata ↓

Abstract

We prove that if the smallest modulus of a covering system is a prime $q \ge 11$, 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.

Explore related subjects

math.NT

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Michael Filaseta, Alexandros Kalogirou, Jonah Klein. 2026-10-25. Results for distinct covering systems with a prime minimum modulus $q \ge 11$. https://arxiv.org/abs/2610.34541

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

On group rings of the simple group of order 168, 504 or 360 and their modules

Let $p$ be a prime and $\mathbb{Z}_p$ the ring of $p$-adic integers. Let $G$ denote the simple group of order 168, 504 or 360. In this paper, we study the structure of the $χ$-part of a $\mathbb{Z}_p[\mathrm{Im}(χ)][G]$-module come from ideal class groups, Artin $L$-functions and Iwasawa theory. After this manuscript was posted, it was pointed out to the author that the main theorem holds for an arbitrary finite group and follows directly from the structure theorem for group rings; see the note added after Theorem 1.1.

math.NT↗
arXiv2026

The $ρ$-Fourier transform

Let $G$ be a reductive group over a local field $F$ and let $ρ:{}^LG \to \mathrm{GL}_{V_ρ}(\mathbb{C})$ be a representation of its $L$-group satisfying suitable assumptions. Braverman, Kazhdan and Ngô conjectured that one has a $ρ$-Fourier transform satisfying certain desiderata. We give a uniform construction of the Fourier transform for arbitrary local fields. This complements work of L.~Lafforgue in the non-Archimedean case, and proves a portion of the conjectures of Braverman, Kazhdan and Ngô. Our methods are spectral in nature.

math.NT↗
arXiv2026

The Hasse principle for diagonal forms restricted to a hypersurface of adjacent degree

We investigate the Hasse principle for Diophantine systems consisting of one diagonal form of degree $k$ and one general form of degree $k-1$. By refining the method of Brandes and Parsell (arXiv:2003.04350) in this specific setting, we improve the bound $n > 2^k k$ to $n > 2^{k-1}(2k-1)$; in particular, the requirement $n > 24$ in the case of degrees three and two is relaxed to $n > 20$.

math.NT↗