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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 set of asymptotic density zero. This gives a negative answer to a conjecture of P. Erd\H os. We pose several problems and a conjecture for further research., Comment: 29pages

For $\alpha >0$, let $$\mathscr{A}=\{ a_1(\log m)^\alpha $ for infinitely many positive integers $m$ and $\ell_m<0.9\log\log m$ for sufficiently integers $m$. Suppose further that $(\ell_i,a_i)=1$ for all $i$. For any $n$, let $f_{\mathscr{A},\mathscr{L}}(n)$ be the number of the available representations listed below $$\ell_in=p+a_i \quad \left(1\le i\le \mathscr{A}(n)\right),$$ where $p$ is a prime number. It is proved that $$\limsup_{n\to \infty } \frac{f_{\mathscr{A},\mathscr{L}}(n)}{\log\log n}>0,$$ which covers an old result of Erd\H os in 1950 by taking $a_i=2^i$ and $\ell_i=1$. One key ingredient in the argument is a technical lemma established here which illustrates how to pick out the admissible parts of an arbitrarily given set of distinct linear functions. The proof then reduces to the verifications of a hypothesis involving well--distributed sets introduced by Maynard, which of course would be the other key ingredient in the argument.

Denote by $\mathcal{R}_p$ the set of all quadratic residues in $\mathbf{F}_p$ for each prime $p$. A conjecture of A. S\'ark\"ozy asserts, for all sufficiently large $p$, that no subsets $\mathcal{A},\mathcal{B}\subseteq\mathbf{F}_p$ with $|\mathcal{A}|,|\mathcal{B}|\geqslant2$ satisfy $\mathcal{A}+\mathcal{B}=\mathcal{R}_p$. In this paper, we show that if such subsets $\mathcal{A},\mathcal{B}$ do exist, then there are at least $(\log 2)^{-1}\sqrt p-1.6$ elements in $\mathcal{A}+\mathcal{B}$ that have unique representations and one should have \begin{align*} \frac{1}{4}\sqrt{p}< |\mathcal{A}|,|\mathcal{B}|< 2\sqrt{p}-1. \end{align*} This refines previous bounds obtained by I.E. Shparlinski, I.D. Shkredov, and Y.-G. Chen and X.-H. Yan. Moreover, we also establish bounds for $|\mathcal{A}|,|\mathcal{B}|$ and the additive energy $E(\mathcal{A},\mathcal{B})$ if few elements in $\mathcal{A}+\mathcal{B}$ have unique representations.

Let $H_n$ be the $n$-th harmonic number and let $v_n$ be its denominator. It is well known that $v_n$ is even for every integer $n\ge 2$. In this paper, we study the properties of $v_n$. One of our results is: the set of positive integers $n$ such that $v_n$ is divisible by the least common multiple of $1, 2, \cdots, \lfloor {n^{1/4}}\rfloor $ has density one. In particular, for any positive integer $m$, the set of positive integers $n$ such that $v_n$ is divisible by $m$ has density one., Comment: 6 pages

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 no proper subset of $A$ is an asymptotic basis of order $h$. In this paper, we resolve a problem of Nathanson on minimal asymptotic bases of order $h$., Comment: 8 pages

In this paper, we consider the equations involving Euler's totient function $\phi$ and Lucas type sequences. In particular, we prove that the equation $\phi (x^m-y^m)=x^n-y^n$ has no solutions in positive integers $x, y, m, n$ except for the trivial solutions $(x, y, m , n)=(a+1, a, 1, 1)$, where $a$ is a positive integer, and the equation $\phi ((x^m-y^m)/(x-y))=(x^n-y^n)/(x-y)$ has no solutions in positive integers $x, y, m, n$ except for the trivial solutions $(x, y, m , n)=(a, b, 1, 1)$, where $a, b$ are integers with $a>b\ge 1$., Comment: 40 pages

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 to $x$. In this paper, we prove that, if $r(A, n)\not= 1$ for all sufficiently large integers $n$, then $|A(x)|> \frac 12 (\log x/\log\log x)^2$ for all sufficiently large $x$., Comment: 10 pages

Recently, motivated by Stanley sequences, Kiss, S\' andor and Yang introduced a new type sequence: a sequence $A$ of nonnegative integers is called an $AP_k$ - covering sequence if there exists an integer $n_0$ such that if $n > n_0$, then there exist $a_1\in A, \dots , a_{k-1}\in A$, $a_1

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 formula for the square-root function. Many mathematicians have paid much attention to congruences on some special colored partition functions. In this paper, we investigate the general colored partition functions. Given positive integers $1=s_1

A positive integer $n$ is called an abundant number if $\sigma (n)\ge 2n$, where $\sigma (n)$ 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 that there are two positive constants $c_1$ and $c_2$ such that $c_1\log\log\log x\le E(x)\le c_2\log\log\log x$. In this paper, we resolve this old problem by proving that, $E(x)/\log \log\log x$ tends to a limit as $x\to +\infty$, and the limit value has an explicit form which is between $3$ and $4$., Comment: 14pages

We present a versatile construction allowing one to obtain pairs of integer sets with infinite symmetric difference, infinite intersection, and identical representation functions.

Let $A=\{a_0,a_1,\ldots,a_{k-1}\}$ be a set of $k$ integers. For any integer $h\ge 1$ and any ordered $k$-tuple of positive integers $\mathbf{r}=(r_0,r_1,\ldots,r_{k-1})$, we define a general $h$-fold sumset, denoted by $h^{(\mathbf{r})}A$, which is the set of all sums of $h$ elements of $A$, where $a_i$ appearing in the sum can be repeated at most $r_i$ times for $i=0,1,\ldots,k-1$. In this paper, we give the best lower bound for $|h^{(\mathbf{r})}A|$ in terms of $\mathbf{r}$ and $h$ and determine the structure of the set $A$ when $|h^{(\mathbf{r})}A|$ is minimal. This generalizes results of Nathanson, and recent results of Mistri and Pandey and also solves a problem of Mistri and Pandey., Comment: 18 pages

In this paper, the following results are proved: (i) For any odd integer $\ell$ with at most two distinct prime factors and any positive integer $n$, the product $(1^\ell+1)(2^\ell+1)\cdots (n^\ell +1)$ is not a powerful number; (ii) For any integer $r\ge 1$, there exists a positive integer $T_r$ such that, if $\ell$ is a positive odd integer with at most $r$ distinct prime factors and $n$ is an integer with $n\ge T_r$, then $(1^\ell+1)(2^\ell+1)\cdots (n^\ell +1)$ is not a powerful number., Comment: 13pages

Let $G$ be a simple connected graph on $n$ vertices and $m$ edges. In [Linear Algebra Appl. 435 (2011) 2570-2584], Lima et al. posed the following conjecture on the least eigenvalue $q_n(G)$ of the signless Laplacian of $G$: $\displaystyle q_n(G)\ge {2m}/{(n-1)}-n+2$. In this paper we prove a stronger result: For any graph with $n$ vertices and $m$ edges, we have $\displaystyle q_n(G)\ge {2m}/{(n-2)}-n+1 (n\ge 6)$.

Let $m$, $k_1$, and $k_2$ be three integers with $m\ge 2$. For any set $A\subseteq \mathbb{Z}_m$ and $n\in \mathbb{Z}_m$, let $\hat{r}_{k_1,k_2}(A,n)$ denote the number of solutions of the equation $n=k_1a_1+k_2a_2$ with $a_1,a_2\in A$. In this paper, using exponential sums, we characterize all $m$, $k_1$, $k_2$, and $A$ for which $\hat{r}_{k_1,k_2}(A,n)=\hat{r}_{k_1,k_2}(\mathbb{Z}_m\setminus A,n)$ for all $n\in \mathbb{Z}_m$. We also pose several problems for further research., Comment: 9 pages

Let $s\ge 2$ be an integer. Denote by $\mu_s$ the least integer so that every integer $\ell >\mu_s$ is the sum of exactly $s$ integers $>1 $ which are pairwise relatively prime. In 1964, Sierpi\'nski asked a determination of $\mu_s$. Let $p_1=2$, $p_2=3, ...$ be the sequence of consecutive primes and let $\mu_s = p_2+p_3+...+p_{s+1}+c_s$. P. Erd\H os proved that there exists an absolute constant $C$ with $-2\le c_s\le C$. In this paper, we determine $\mu_s$ for all $s\ge 2$. As a corollary, we show that $-2\le c_s\le 1100$ and the set of integers $s$ with $\mu_s= p_2+p_3+... +p_{s+1}+1100$ has the asymptotic density 1., Comment: 15pages

In 1946, P. Erd\H os and I. Niven proved that there are only finitely many positive integers $n$ for which one or more elementary symmetric functions of $1, 1/2, \ldots , 1/n$ are integers. In this paper we solve this old problem by showing that if $n\ge 4$, then none of elementary symmetric functions of $1, 1/2, \ldots , 1/n$ is an integer., Comment: 7pages

Let $T_o(k)$ denote the number of solutions of $\sum_{i=1}^k\frac 1{x_i}=1$ in odd numbers $11$ in $S(p_1, p_2,..., p_t)$ is more than 1. In this paper we study $T_o(k)$ and $T_k(p_1,..., p_t)$. Three of our results are: 1) $T_o(2k+1)\ge (\sqrt 2)^{(k+1)(k-4)}$ for all $k\ge 4$; 2) if the inverse sum of all elements $s_j>1$ in $S(p_1, p_2,..., p_t)$ is more than 1, then $T_k(p_1,..., p_t)\not= 0$ for infinitely many $k$ and the set of these $k$ is the union of finitely many arithmetic progressions; 3) there exists two constants $k_0=k_0(p_1,..., p_t)>1$ and $c=c(p_1,..., p_t)>1$ such that for any $k>k_0$ we have either $T_k(p_1,..., p_t)= 0$ or $T_k(p_1,..., p_t)>c^k$., Comment: 18pages

Let $p_1, p_2,..., p_n$ be distinct primes. In 1970, Erd\H os, Herzog and Sch\"{o}nheim proved that if $\cal D$ is a set of divisors of $N=p_1^{\alpha_1}...p_n^{\alpha_n}$, $\alpha_1\ge \alpha_2\ge...\ge \alpha_n$, no two members of the set being coprime and if no additional member may be included in $\cal D$ without contradicting this requirement then $ |{\cal D}|\ge \alpha_n \prod_{i=1}^{n-1} (\alpha_i +1)$. They asked to determine all sets $\cal D$ such that the equality holds. In this paper we solve this problem. We also pose several open problems for further research., Comment: 12 pages

Let $f(n,k)$ be the largest number of positive integers not exceeding $n$ from which one cannot select $k+1$ pairwise coprime integers, and let $E(n,k)$ be the set of positive integers which do not exceed $n$ and can be divided by at least one of $p_1, p_2,..., p_k$, where $p_i$ is the $i$-th prime. In 1962, P. Erd\H os conjectured that $f(n,k)=|E(n,k)|$ for all $n\ge p_k$. In 1973, S. L. G. Choi proved that the conjecture is true for $k=3$. In 1994, Ahlswede and Kachatrian disproved the conjecture for $k=212$. In this paper we prove that, for $n\ge 49$, if A(n,4) is a set of positive integers not exceeding $n$ from which one cannot select 5 pairwise coprime integers and $|A(n,4)|\ge |E(n,4)|$, then $A(n,4)=E(n,4)$. In particular, the conjecture is true for k=4. Several open problems and conjectures are posed for further research., Comment: 17 pages

The Schinzel hypothesis claims (but it seems hopeless to prove) that any irreducible Q[x] polynomial without a constant factor assumes infinitely many prime values at integer places. On the other hand, it is easy to see that a reducible Q[x] polynomial can have only finitely many such places. In this paper we prove that a reducible Z[x] polynomial of degree n (where n is not 4 or 5) can have at most n+2 such places, and there exist examples with n+1 places. If the Schinzel hypothesis and the k-prime-tuple conjecture are true, then there are also polynomials with n+2 such places. (For n=4 or 5 the maximum possible value is 8.) If a Z[x] polynomial is the product of two non-constant integer-valued Q[x] polynomials, then there are at most 1.87234...n+o(n) such places. Even in this case, the number of integer places with positive prime values is at most n. More generally, if f=gh \in R[x] is a product of two non-constant real polynomials, then the number of real places x such that |g(x)|=1 or |h(x)=1|, while f(x)>1, is at most n. For a natural complex version of this last statement we give a counterexample. Furthermore, we briefly consider the generalized problem of assuming values in a fixed finite set of integers., Comment: 12 pages