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Let $a$ and $b$ be positive integers. In 1946, ErdH{o}s and Niven proved that there are only finitely many positive integers $n$ for which one or more of the elementary symmetric functions of $1/b, 1/(a+b),..., 1/(an-a+b)$ are integers. In this paper, we show that for any integer $k$ with $1le kle n$, the $k$-th elementary symmetric function of $1/b, 1/(a+b),..., 1/(an-a+b)$ is not an integer except that either $b=n=k=1$ and $age 1$, or $a=b=1, n=3$ and $k=2$. This refines the ErdH{o}s-Niven theorem and answers an open problem raised by Chen and Tang in 2012.
Given an abelian group $G$, it is natural to ask whether there exists a permutation $pi$ of $G$ that destroys all nontrivial 3-term arithmetic progressions (APs), in the sense that $pi(b) - pi(a) eq pi(c) - pi(b)$ for every ordered triple $(a,b,c) i
Erd{o}s and Niven proved in 1946 that for any positive integers $m$ and $d$, there are at most finitely many integers $n$ for which at least one of the elementary symmetric functions of $1/m, 1/(m+d), ..., 1/(m+(n-1)d)$ are integers. Recently, Wang a
We prove a result on the distribution of the general divisor functions in arithmetic progressions to smooth moduli which exceed the square root of the length.
We study a Dirichlet series in two variables which counts primitive three-term arithmetic progressions of squares. We show that this multiple Dirichlet series has meromorphic continuation to $mathbb{C}^2$ and use Tauberian methods to obtain counts fo
For relatively prime positive integers $u_0$ and $r$, we consider the least common multiple $L_n:=mathrm{lcm}(u_0,u_1,ldots, u_n)$ of the finite arithmetic progression ${u_k:=u_0+kr}_{k=0}^n$. We derive new lower bounds on $L_n$ which improve upon th