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 theelementary symmetric functions of $1/m, 1/(m+d), ..., 1/(m+(n-1)d)$ areintegers. Recently, Wang and Hong refined this result by showing that if $n\geq4$, 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 apolynomial of degree at least $2$ and of nonnegative integer coefficients. Inthis 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 $m\geq2$ beingan integer and $n=1$.
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机译:Erd \“ {o} s和Niven在1946年证明,对于任何正整数$ m $和$ d $,最多有有限个整数$ n $,对于其中至少一个基本对称函数$ 1 / m,1 /(m + d),...,1 /(m +(n-1)d)$是整数。最近,Wang和Hong通过证明如果$ n \ geq4 $,则没有一个基本对称函数来精炼此结果。 $ 1 / m,1 /(m + d),...,1 /(m +(n-1)d)$的整数是任何正整数$ m $和$ d $的整数。设$ f $为多项式的度数至少为$ 2 $和非负整数系数。在本文中,我们证明没有$ 1 / f(1),1 / f(2),...,1 / f(n)$的基本对称函数是除$ f(x)= x ^ {m} $以外的整数,其中$ m \ geq2 $是整数,$ n = 1 $。
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