  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.
