Related papers: Zhi-Wei Sun's 1-3-5 Conjecture and Variations

We prove two explicit formulae for one-part double Hurwitz numbers with completed 3-cycles. We define "combinatorial Hodge integrals" from these numbers in the spirit of the celebrated ELSV formula. The obtained results imply some explicit…

The Ratios Conjecture of Conrey, Farmer and Zirnbauer predicts the answers to numerous questions in number theory, ranging from n-level densities and correlations to mollifiers to moments and vanishing at the central point. The conjecture…

We show that there are $O(B^{3/5-3/1555+\ep})$ triples $(x,y,z)$ of square-full integesr up to $B$ satisfying the equation $x+y=z$ for any fixed $\ep>0$. This is the first improvement over the `easy' exponent $3/5$, given by Browning and…

Let $E_n$ be the $n$-th Euler number and $(a)_n=a(a+1)\cdots (a+n-1)$ the rising factorial. Let $p>3$ be a prime. In 2012, Sun proved the that $$ \sum^{(p-1)/2}_{k=0}(-1)^k(4k+1)\frac{(\frac{1}{2})_k^3}{k!^3} \equiv…

It is well known that the following Collatz Conjecture is one of the unsolved problems in mathematics. Collatz Conjecture: For any positive integer $n>1$, the following recursive algorithm will convergent to 1 by a finite number of steps.…

Let $p>3$ be a prime number. We compute the rings of invariants of the elementary abelian $p$-group $(\mathbb Z/p\mathbb Z)^r$ for $3$-dimensional generic representations. Furthermore we show that these rings of invariants are complete…

We study matrix forms of quaternionic versions of the Fourier Transform and Convolution operations. Quaternions offer a powerful representation unit, however they are related to difficulties in their use that stem foremost from…

We prove a conjecture that classifies exceptional numbers. This conjecture arises in two different ways, from cryptography and from coding theory. An odd integer $t\geq 3$ is said to be exceptional if $f(x)=x^t$ is APN (Almost Perfect…

In this short note, we give an affirmative answer to Wu's conjecture on practical numbers, which was posed in [X.-H. Wu, {\it Special forms and the distribution of practical numbers}, Acta Math. Hungar., {\bf 160}(2020), 405-411].

In 1961, P. Erd\H{o}s, A. Ginzburg, and A. Ziv proved a remarkable theorem stating that each set of $2n-1$ integers contains a subset of size $n$, the sum of whose elements is divisible by $n$. We will prove a similar result for pairs of…

This article examines the nontrivial solutions of the congruence \[ (p-1)\cdots(p-r) \equiv -1 \pmod p. \] We discuss heuristics for the proportion of primes $p$ that have exactly $N$ solutions to this congruence. We supply numerical…

A classic conjecture of F\"{u}redi, Kahn and Seymour (1993) states that given any hypergraph with non-negative edge weights $w(e)$, there exists a matching $M$ such that $\sum_{e \in M} (|e|-1+1/|e|)\, w(e) \geq w^*$, where $w^*$ is the…

The generalized Erd\H{o}s-Straus conjecture, proposed by Wac\l{}aw Sierpi\'{n}ski in 1956, asks whether the Diophantine equation \[ \frac{5}{a} = \frac{1}{b} + \frac{1}{c} + \frac{1}{d} \] admits positive integer solutions $b,c,d \in…

We prove versions of Khintchine's Theorem (1924) for approximations by rational numbers whose numerators lie in randomly chosen sets of integers, and we explore the extent to which the monotonicity assumption can be removed. Roughly…

We establish some supercongruences related to a supercongruence of Van Hamme, such as \begin{align*} \sum_{k=0}^{(p+1)/2} (-1)^k (4k-1)\frac{(-\frac{1}{2})_k^3}{k!^3} &\equiv p(-1)^{(p+1)/2}+p^3(2-E_{p-3})\pmod{p^{4}},\\…

We show the integrality of the simple Hurwitz numbers. The main tool is the cut-and-join operator, and our proof is a purely combinatorial one.

Consider a finite positive integer. If it is even, divide it by 2, and if it is odd, multiply it by 3 and add 1. This will give you a new integer. Following the procedure for the new integer, you will receive another integer. Repeat the…

We prove that every subset of $\{1,\dots, N\}$ which does not contain any solutions to the equation $x+y+z=3w$ has at most $\exp(-c(\log N)^{1/5+o(1)})N$ elements, for some $c>0$. This theorem improves upon previous estimates. Additionally,…

Hadwiger's conjecture implies that $n\le\alpha h$ for all graphs of order $n$, stability number $\alpha $, and Hadwiger number $h$. Combining ideas of Kawarabayashi et al. and Wood, we prove that $n\le (\alpha-1)(2h-5)+5$ for such graphs if…

We consider a variety of Euler's conjecture, i.e., whether the Diophantine system \[\begin{cases} n=a_{1}+a_{2}+\cdots+a_{s-1}, a_{1}a_{2}\cdots a_{s-1}(a_{1}+a_{2}+\cdots+a_{s-1})=b^{s} \end{cases}\] has solutions…