Song Guo, Zhi-Wei Sun
A famous unsolved conjecture of P. Erdos and J. L. Selfridge states that there does not exist a covering system {a_s(mod n_s)}_{s=1}^k with the moduli n_1,...,n_k odd, distinct and greater than one. In this paper we show that if such a covering system {a_s(mod n_s)}_{s=1}^k exists with n_1,...,n_k all square-free, then the least common multiple of n_1,...,n_k has at least 22 prime divisors.
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On the $j$-th smallest modulus of a covering system with distinct moduli
本研究在Balister等人对Hough关于不同模覆盖系统最小模有界性的证明基础上，推导出具有不同模的极小覆盖系统的第$j$小模的上界为$\exp(cj^2/\log(j+1))$，其中$c$为正绝对常数。
We generalise Uspensky's theorem characterising eventual exact (e.e.) covers of the positive integers by homogeneous Beatty sequences, to e.e. m-covers, for any m \in \N, by homogeneous sequences with irrational moduli. We also consider inhomogeneous sequences, again with irrational moduli, and obtain a purely arithmetical characterisation of e.e. m-covers. This generalises a result of Graham for m = 1, but when m > 1 the arithmetical description is more complicated. Finally we speculate on how one might make sense of the notion of an exact m-cover when m is not an integer, and present a "frac…
A covering system is a finite collection of arithmetic progressions whose union is the set of integers. The study of covering systems with distinct moduli was initiated by Erdős in 1950, and over the following decades numerous problems were posed regarding their properties. One particularly notorious question, due to Erdős, asks whether there exist covering systems whose moduli are distinct and all odd. We show that if in addition one assumes the moduli are square-free, then there must be an even modulus.
In this paper we construct a cover {a_s(mod n_s)}_{s=1}^k of Z with odd moduli such that there are distinct primes p_1,...,p_k dividing 2^{n_1}-1,...,2^{n_k}-1 respectively. Using this cover we show that for any positive integer m divisible by none of 3, 5, 7, 11, 13 there exists an infinite arithmetic progression of positive odd integers the m-th powers of whose terms are never of the form $2^n\pm p^a$ with p a prime and a,n in {0,1,2,...}. We also construct another cover of Z with odd moduli and use it to prove that $x^2-F_{3n}/2$ has at least two distinct prime factors whenever n is a nonne…
Based on work of P. Balister, B. Bollobás, R. Morris, J. Sahasrabudhe and M. Tiba, we show that if a covering system has distinct squarefree moduli, then the minimum modulus is at most 118. We also show that in general the $k^{\rm th}$ smallest modulus in a covering system with distinct moduli (provided it is required for the covering) is bounded by an absolute constant.
Covering systems with odd moduli
本文研究奇数覆盖问题的一个变体，允许一个奇素数模重复出现而其余模互异且为奇数，并进一步考虑模为无平方因子的情形。
A further investigation on covering systems with odd moduli
本文研究一种奇数覆盖问题变体，允许一个奇数模多次出现，其余模为不同大于1的奇数。
Covering systems with the sum of the reciprocals of the moduli close to $1$
本文证明了在有限不同模覆盖系统中，当最小模数大于4时，模数倒数之和必远离1，解决了Davenport提出的问题并证实了Erdos和Selfridge的猜想。
On the intervals for the non-existence of covering systems with distinct moduli
本研究在数论领域的覆盖系问题上，基于前人方法进一步拓展区间边界，证明对任意整数n≥3，不存在所有模均落在[n,10n]区间内的不同模覆盖系，推进了相关结论的区间上界。
Since their introduction by Erdős in 1950, covering systems (that is, finite collections of arithmetic progressions that cover the integers) have been extensively studied, and numerous questions and conjectures have been posed regarding the existence of covering systems with various properties. In particular, Erdős asked if the moduli can be distinct and all arbitrarily large, Erdős and Selfridge asked if the moduli can be distinct and all odd, and Schinzel conjectured that in any covering system there exists a pair of moduli, one of which divides the other. Another beautiful conjecture, propo…
Covering systems where the prime divisors of all moduli are only $2$, $3$, or $5$
本文研究了所有正整数四元组(m,a,b,c)的问题，其中a≥b≥c，使得存在一个具有最小模m和模数的最小公倍数为2^a 3^b 5^c的覆盖系统。当m=2,3,4,5或6时，除了m=6且b=c=1的情况外，得到了所有这样的四元组的完整描述。还证明了如果模数的最小公倍数仅包含2，3或5作为质因数，则m≤9，并构造了一个具有m=8，a=8，b=3，c=2的覆盖系统。当存在覆盖系统时，提供了示例。通过整数规划或新的估计方法证明了覆盖系统的不存在性。
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.
Computations and observations on congruence covering systems
该研究针对同余覆盖系领域，完成了模数不超过10的不同模覆盖系的完整分类，并据此得出所有模大于2的不同模覆盖系的最小基数为11的结论。
A set of arithmetical sequences $$ a_1\, (\bmod{ \,\, m_1}) \quad, \quad a_2 \, (\bmod{\,\, m_2}) \quad, \quad \dots \quad , \quad a_k \, (\bmod{\,\,m_k}) \quad \quad , $$ with $$ m_1 \leq m_2 \leq \dots \leq m_k \quad \quad , $$ is called a {\it disjoint covering system} (alias {\it exact covering system}) if every positive integer belongs to {\bf exactly} one of the sequences. Mirski, Newman, Davenport and Rado famously proved that the moduli can't all be distinct. In fact the two largest moduli must be equal, i.e. $m_{k-1}=m_k$ This raises the natural question:"How close can you get to gett…
A covering system of the integers is a finite collection of modular residue classes $\{a_m \bmod{m}\}_{m \in S}$ whose union is all integers. Given a finite set $S$ of moduli, it is often difficult to tell whether there is a choice of residues modulo elements of $S$ covering the integers. Hough has shown that if the smallest modulus in $S$ is at least $10^{16}$, then there is none. However, the question of whether there is a covering of the integers with all odd moduli remains open. We consider multiplicative restrictions on the set of moduli to generalize Hough's negative solution to the mini…
Kernel-Checked Exclusions for the Erdős-Selfridge Odd Covering Problem: Any Odd Covering of $\mathbb{Z}$ Has lcm Exceeding 10000
研究埃尔德什 - 塞尔弗里奇奇数覆盖问题，通过Lean 4形式化证明，利用密度论证、丰富度下限、中国剩余容量证书及枚举等方法，得出整数集\(\mathbb{Z}\)的奇数覆盖其模最小公倍数超10000的结论，贡献在于提供内核检查的认知成果。
On a conjecture of Krukenberg and a problem of Dalton and Trifonov
本文证明了不同模数覆盖系在最小模数为5时最大模数≥108、最小公倍数≥1440，最小模数为6时最小公倍数≥5040（均为最优值），解决了Krukenberg猜想及Dalton与Trifonov的问题。
A positive integer n is called a covering number if there are some distinct divisors n_1,...,n_k of n greater than one and some integers a_1,...,a_k such that Z is the union of the residue classes a_1(mod n_1),...,a_k(mod n_k). A covering number is said to be primitive if none of its proper divisors is a covering number. In this paper we give some sufficient conditions for n to be a (primitive) covering number; in particular, we show that for any r=2,3,... there are infinitely many primitive covering numbers having exactly r distinct prime divisors. In 1980 P. Erdos asked whether there are inf…
In 2007, Zhi-Wei Sun defined a \emph{covering number} to be a positive integer $L$ such that there exists a covering system of the integers where the moduli are distinct divisors of $L$ greater than 1. A covering number $L$ is called \emph{primitive} if no proper divisor of $L$ is a covering number. Sun constructed an infinite set $\mathcal L$ of primitive covering numbers, and he conjectured that every primitive covering number must satisfy a certain condition. In this paper, for a given $L\in \mathcal L$, we derive a formula that gives the exact number of coverings that have $L$ as the least…
Let {a_s(mod n_s)}_{s=1}^k (k>1) be a finite system of residue classes with the moduli n_1,...,n_k distinct. By means of algebraic integers we show that the range of the covering function w(x)=|{1\le s\le k: x=a_s (mod n_s)}| is not contained in any residue class with modulus greater one. In particular, the values of w(x) cannot have the same parity.
On covering systems of polynomial rings over finite fields
本文基于扭曲方法，证明了有限域上多项式环覆盖系统的模数最小次数有界于s和q依赖的常数，解决了有限域上的最小模问题并推翻了Azlin的猜想。
Emmanuel Kowalski and William Sawin proved, using a deep independence result of Kloosterman sheaves, that the polygonal paths joining the partial sums of the normalized classical Kloosterman sums S(a,b0;p)/p^{1/2} converge in the sense of finite distributions to a specific random Fourier series, as a varies over (Z/pZ)^*, b0 is fixed in (Z/pz)* and p tends to infinity among the odd prime numbers. This article considers the case of S(a,b0;p^n)/p^{n/2}, as a varies over (Z/p^nZ)^*, b0 is fixed in (Z/p^nZ)^*, p tends to infinity among the odd prime numbers and n>=2 is a fixed integer. A convergen…
On covering radius of generalized Zetterberg codes
本研究运用特征和与Weil型估计的解析数论方法，研究有限域上广义Zetterberg码的覆盖半径，证明其至多为3，确定了多参数下精确值，填补偶特征域研究空白，拓宽奇特征域参数范围，得到无穷多拟完美和极大码。
On the minimum modulus problem in number fields
本文通过引入逆系统相关概率测度理论发展畸变方法，解决了数域中的Erdős最小模问题，还证明了数域极小覆盖系中第$j$个最小范数有界。
Heights and singular moduli of Drinfeld modules
本文发展了Drinfeld模的高技术，证明了在$q>5$为奇数时，秩为2的$\mathbb{F}_q[t]$-Drinfeld模中至多仅有有限个奇异模为代数单位。
Bounds for Erdős covering systems in global function fields
本文研究了全球函数域中覆盖系统的界，证明了在特定条件下不存在具有乘法次数s的覆盖系统，并改进了之前与李、王和伊合作的研究结果。
We show that the classical Kuga-Satake construction gives rise, away from characteristic 2, to an open immersion from the moduli of primitively polarized K3 surfaces (of any fixed degree) to a certain regular integral model for a Shimura variety of orthogonal type. This allows us to attach to every polarized K3 surface in odd characteristic an abelian variety such that divisors on the surface can be identified with certain endomorphisms of the attached abelian variety. In turn, this reduces the Tate conjecture for K3 surfaces over finitely generated fields of odd characteristic to a version of…
On Erdős covering systems in global function fields
本文基于 distortion 方法，证明了在特定条件下，全球函数域中不存在具有特定乘数的覆盖系统，尤其指出当 q≥759 时，F_q[x] 上不存在具有不同模数的覆盖系统。
To every covering of curves, we associate several varieties having the same field of moduli and same fields of definition. We deduce examples of curves having Q (the field of rationals) as field of moduli, that admit models over any completion of Q but no model over Q.
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