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
nd Hong refined this result by showing that if $ngeq 4$, then none of the elementary symmetric functions of $1/m, 1/(m+d), ..., 1/(m+(n-1)d)$ is an integer for any positive integers $m$ and $d$. Let $f$ be a polynomial of degree at least $2$ and of nonnegative integer coefficients. In this paper, we show that none of the elementary symmetric functions of $1/f(1), 1/f(2), ..., 1/f(n)$ is an integer except for $f(x)=x^{m}$ with $mgeq2$ being an integer and $n=1$.
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.
Let $n$ be a positive integer and $f(x)$ be a polynomial with nonnegative integer coefficients. We prove that ${rm lcm}_{lceil n/2rceil le ile n} {f(i)}ge 2^n$ except that $f(x)=x$ and $n=1, 2, 3, 4, 6$ and that $f(x)=x^s$ with $sge 2$ being an integ
er and $n=1$, where $lceil n/2rceil$ denotes the smallest integer which is not less than $n/2$. This improves and extends the lower bounds obtained by Nair in 1982, Farhi in 2007 and Oon in 2013.
In this paper, we determine all the squares in the sequence ${prod_{k=2}^n(k^2-1)}_{n=2}^infty $. From this, one deduces that there are infinitely many squares in this sequence. We also give a formula for the $p$-adic valuation of the terms in this sequence.
