arXiv subjects

Jack Dalton

Publications and source records attributed to Jack Dalton.

4 recordsLinked to original sources

arXiv2025

On the intervals for the non-existence of covering systems with distinct moduli

In a research seminar in $2006$, M. Filaseta, O. Trifonov, and G. Yu showed for each integer $n\geq3$ there is no distinct covering with all moduli in the interval $[n, 6n]$. In $2022$, this interval was subsequently improved to $[n, 8n]$ by the first author and O. Trifonov. The first author then improved this bound to $[n, 9n]$ in $2023$. Building off their method, we show that for each integer $n\geq 3$, there does not exist a distinct covering system with all moduli in the interval $[n, 10n]$.

math.NT↗

arXiv2023

Lattice points close to a helix

We obtain lower bound for the maximum distance between any three distinct points in an affine lattice which are close to a helix with small curvature and torsion.

math.NT↗

arXiv2023

Representing positive integers as a sum of a squarefree number and a small prime

We prove that every positive integer $n$ which is not equal to $1$, $2$, $3$, $6$, $11$, $30$, $155$, or $247$ can be represented as a sum of a squarefree number and a prime not exceeding $\sqrt{n}$.

math.NT↗

arXiv2022

Extreme Covering Systems of the Integers

It is proved that if the least modulus of a distinct covering system is 4, its largest modulus is at least 60; also if the least modulus is 3, the LCM of the moduli is at least 120; finally, if the least modulus is 4, the LCM of the moduli is at least 360. The constants 60, 120, and 360 can not be replaced by larger constants.

math.NT↗