On a conjecture of Krukenberg and a problem of Dalton and Trifonov

TL;DR

This paper proves optimal bounds on the largest modulus and least common multiple in covering systems with minimal modulus 5 or 6, resolving longstanding conjectures and problems in number theory.

Contribution

It establishes the best possible bounds for covering systems with minimal moduli 5 and 6, confirming conjectures by Krukenberg, Dalton, and Trifonov.

Findings

Largest modulus at least 108 when minimal modulus is 5

LCM of moduli at least 1440 when minimal modulus is 5

LCM of moduli at least 5040 when minimal modulus is 6

Abstract

We prove that if the smallest modulus of a covering system with distinct moduli is $5$, then the largest modulus is at least 108. We also prove that if the smallest modulus of a covering system with distinct moduli is $5$, then the least common multiple of the moduli is at least 1440. Finally, we prove that if the smallest modulus of a covering system with distinct moduli is 6, then the least common multiple of the moduli is at least $5040$. The constants $108$, $1440$ and $5040$ are best possible. This resolves a conjecture of Krukenberg, a problem of Dalton and Trifonov, and a generalization thereof.