arXiv subjects

Jonah Klein

Publications and source records attributed to Jonah Klein.

4 recordsLinked to original sources

arXiv2026

Covering systems where the prime divisors of all moduli are only $2$, $3$, or $5$

We try to find all quadruples of positive integers $(m,a,b,c)$ with $a \geq b \geq c$ such that there exists a distinct covering system with minimum modulus $m$ and least common multiple of the moduli $2^a 3^b 5^c$. We obtain complete description of all such quadruples when $m=2,3,4,5$, or $6$, except when $m=6$ and $b=c=1$. We also show that if the LCM of the moduli has only $2$, $3$, or $5$ as prime divisors, then $m \leq 9$ and construct a distinct covering system with $m=8$, $a=8$, $b=3$, and $c=2$. When a covering system exists for a quadruple $(m,a,b,c)$ we provide an example. Nonexistence of covering systems is established via integer programming or by using a new estimate on the density of a set covered by a system of congruences.

math.NT↗

arXiv2025

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

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.

math.NT↗

arXiv2024

On extreme values of $r_3(n)$ in arithmetic progressions

For a given integer $m$ and any residue $a \pmod{m}$ that can be written as a sum of 3 squares modulo $m$, we show the existence of infinitely many integers $n \equiv a \pmod{m}$ such that the number of representations of $n$ as a sum of three squares, $r_3(n)$, satisfies $r_3(n) \gg_m \sqrt{n} \log \log n$. Consequently, we establish that there are infinitely many integers $n \equiv a \pmod{m}$ for which the Hurwitz class number $H(n)$ also satisfies $H(n) \gg_m \sqrt{n} \log \log n$.

math.NT↗

arXiv2022

On the $j$-th smallest modulus of a covering system with distinct moduli

Covering systems were introduced by Erd\H{o}s in 1950. In the same article where he introduced them, he asked if the minimum modulus of a covering system with distinct moduli is bounded. In 2015, Hough answered affirmatively this long standing question. In 2022, Balister, Bollob\'as, Morris, Sahasrabudhe and Tiba gave a simpler and more versatile proof of Hough's result. Building upon their work, we show that there exists some absolute constant $c>0$ such that the $j$-th smallest modulus of a minimal covering system with distinct moduli is $\le \exp(cj^2/\log(j+1))$.

math.NT↗