Yong-Gao Chen
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View article: The number of primes not in a numerical semigroup
The number of primes not in a numerical semigroup Open
For two coprime positive integers $a$ and $b$,let $π^* (a, b)$ be the number of primes that cannot be represented as $au+bv$, where $u$ and $v$ are nonnegative integers. It is clear that $π^* (a, b)\le π(ab-a-b)$, where $π(x)$ denotes the …
View article: On the sum of a prime and two Fibonacci numbers
On the sum of a prime and two Fibonacci numbers Open
Let $\{f_n\}$ be the Fibonacci sequence. For any positive integer $n$, let $r(n)$ be the number of solutions of $n=p+f_{k_1^{2}} +f_{k_{2}^{2}}$, where $p$ is a prime and $k_1, k_2$ are nonnegative integers with $k_1\le k_2$. In this paper…
View article: On subsets of asymptotic bases
On subsets of asymptotic bases Open
Let be an integer. In this paper, we prove that if is an asymptotic basis of order and is a nonempty subset of , then either there exists a finite subset of such that is an asymptotic basis of order , or for any , there exists a fin…
View article: Boric Acid–Driven Surface Charge Reconstruction on Oxygen–Rich Zero–Valent Iron Enhances Uranium Capture Efficiency
Boric Acid–Driven Surface Charge Reconstruction on Oxygen–Rich Zero–Valent Iron Enhances Uranium Capture Efficiency Open
View article: A conjecture of Erdős on $p+2^k$
A conjecture of Erdős on $p+2^k$ Open
Let $\mathcal{U}$ be the set of positive odd integers that cannot be represented as the sum of a prime and a power of two. In this paper, we prove that $\mathcal{U}$ is not a union of finitely many infinite arithmetic progressions and a se…
View article: On infinite arithmetic progressions in sumsets
On infinite arithmetic progressions in sumsets Open
View article: On $d$-complete sequences of integers, II
On $d$-complete sequences of integers, II Open
In 1996, Erdős and Lewin introduced the notion of $d$-complete sequences. A sequence $\mathcal T$ of positive integers is called $d$-complete if every sufficiently large integer can be represented as the sum of distinct terms taken from $\…
View article: On a conjecture of Erdős
On a conjecture of Erdős Open
In this note, we confirm an old conjecture of Erdős.
View article: Quantitative results of the Romanov type representation functions
Quantitative results of the Romanov type representation functions Open
For $α>0$, let $$\mathscr{A}=\{ a_1(\log m)^α$ for infinitely many positive integers $m$ and $\ell_m<0.9\log\log m$ for sufficiently integers $m$. Suppose further that $(\ell_i,a_i)=1$ for all $i$. For any $n$, let $f_{\mathscr{A},\mathscr…
View article: A conjecture of Sárközy on quadratic residues, II
A conjecture of Sárközy on quadratic residues, II Open
Denote by $\mathcal{R}_p$ the set of all quadratic residues in $\mathbf{F}_p$ for each prime $p$. A conjecture of A. Sárközy asserts, for all sufficiently large $p$, that no subsets $\mathcal{A},\mathcal{B}\subseteq\mathbf{F}_p$ with $|\ma…
View article: On a conjecture of Erdős
On a conjecture of Erdős Open
Let $\mathcal{P}$ denote the set of all primes. In 1950, P. Erdős conjectured that if $c$ is an arbitrarily given constant, $x$ is sufficiently large and $a_1,\dots , a_t$ are positive integers with $a_1\log x$, then there exists an intege…
View article: On a conjecture of Erd\\H{o}s
On a conjecture of Erd\\H{o}s Open
Let $\\mathcal{P}$ denote the set of all primes. In 1950, P. Erd\\H{o}s\nconjectured that if $c$ is an arbitrarily given constant, $x$ is sufficiently\nlarge and $a_1,\\dots , a_t$ are positive integers with\n$a_1
View article: Correction to: On positive integers n with σl(2n + 1) < σl(2n)
Correction to: On positive integers n with σl(2n + 1) < σl(2n) Open
View article: Diophantine equations involving Euler’s totient function
Diophantine equations involving Euler’s totient function Open
In this paper, we consider the equations involving Euler's totient function $ϕ$ and Lucas type sequences. In particular, we prove that the equation $ϕ(x^m-y^m)=x^n-y^n$ has no solutions in positive integers $x, y, m, n$ except for the triv…
View article: REPRESENTATION FUNCTIONS ON ABELIAN GROUPS
REPRESENTATION FUNCTIONS ON ABELIAN GROUPS Open
Let $G$ be a finite abelian group, $A$ a nonempty subset of $G$ and $h\geq 2$ an integer. For $g\in G$ , let $R_{A,h}(g)$ denote the number of solutions of the equation $x_{1}+\cdots +x_{h}=g$ with $x_{i}\in A$ for $1\leq i\leq h$ . Kiss e…
View article: On additive representation functions
On additive representation functions Open
Let $A$ be an infinite set of natural numbers. For $n\in \mathbb{N}$, let $r(A, n)$ denote the number of solutions of the equation $n=a+b$ with $a, b\in A, a\le b$. Let $|A(x)|$ be the number of integers in $A$ which are less than or equal…
View article: Corrigendum to “Non-Wieferich primes in arithmetic progressions”
Corrigendum to “Non-Wieferich primes in arithmetic progressions” Open
View article: On the denominators of harmonic numbers
On the denominators of harmonic numbers Open
Let be the n -th harmonic number and let be its denominator. It is well known that is even for every integer . In this paper, we study the properties of . One of our results is: the set of positive integers n such that is divisible by …
View article: On AP3-covering sequences
On AP3-covering sequences Open
Recently, motivated by Stanley's sequences, Kiss, Sándor, and Yang introduced a new type sequence: a sequence A of nonnegative integers is called an -covering sequence if there exists an integer such that, if , then there exist , such th…
View article: On a problem of Nathanson
On a problem of Nathanson Open
A set $A$ of nonnegative integers is an asymptotic basis of order $h$ if every sufficiently large integer can be represented as the sum of $h$ integers (not necessarily distinct) of $A$. An asymptotic basis $A$ of order $h$ is minimal if n…
View article: ON ADDITIVE REPRESENTATION FUNCTIONS
ON ADDITIVE REPRESENTATION FUNCTIONS Open
For any finite abelian group $G$ with $|G|=m$ , $A\subseteq G$ and $g\in G$ , let $R_{A}(g)$ be the number of solutions of the equation $g=a+b$ , $a,b\in A$ . Recently, Sándor and Yang [‘A lower bound of Ruzsa’s number related to the Erdős…
View article: JEŚMANOWICZ’ CONJECTURE ON PYTHAGOREAN TRIPLES
JEŚMANOWICZ’ CONJECTURE ON PYTHAGOREAN TRIPLES Open
In 1956, Jeśmanowicz conjectured that, for any positive integers $m$ and $n$ with $m>n$ , $\gcd (m,n)=1$ and $2\nmid m+n$ , the Diophantine equation $(m^{2}-n^{2})^{x}+(2mn)^{y}=(m^{2}+n^{2})^{z}$ has only the positive integer solution $(x…
View article: Asymptotic formulas for general colored partition functions
Asymptotic formulas for general colored partition functions Open
In 1917, Hardy and Ramanujan obtained the asymptotic formula for the classical partition function $p(n)$. The classical partition function $p(n)$ has been extensively studied. Recently, Luca and Ralaivaosaona obtained the asymptotic formul…
View article: Integer Sets with Identical Representation Functions
Integer Sets with Identical Representation Functions Open
See the abstract in the attached pdf.
View article: On the $r$-th Root Partition Function
On the $r$-th Root Partition Function Open
The well known partition function $p(n)$ has a long research history, where $p(n)$\ndenotes the number of solutions of the equation $n = a_1 + \\cdots + a_k$ with\nintegers $1 \\leq a_1 \\leq \\cdots \\leq a_k$. In this paper, we investiga…
View article: Non-Wieferich primes in arithmetic progressions
Non-Wieferich primes in arithmetic progressions Open
Graves and Murty proved that for any integer $a\ge 2$ and any fixed integer $k\ge 2$, there are $\gg \log x/\log \log x$ primes $p\le x$ such that $a^{p-1}\not \equiv 1\pmod {p^2}$ and $p\equiv 1\pmod k$, under the assumption of the abc co…
View article: On consecutive abundant numbers
On consecutive abundant numbers Open
A positive integer $n$ is called an abundant number if $σ(n)\ge 2n$, where $σ(n)$ is the sum of all positive divisors of $n$. Let $E(x)$ be the largest number of consecutive abundant numbers not exceeding $x$. In 1935, P. Erd\H os proved t…
View article: On the cardinality of general <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" display="inline" overflow="scroll"><mml:mi>h</mml:mi></mml:math>-fold sumsets
On the cardinality of general -fold sumsets Open
View article: On a conjecture of Sárközy and Szemerédi
On a conjecture of Sárközy and Szemerédi Open
Two infinite sequences $A$ and $B$ of non-negative integers are called infinite additive complements if their sum contains all sufficiently large integers. In 1994, Sárközy and Szemerédi conjectured that there exist infinite additiv…