For
, it is trivially true. Let
.
It is enough to show that the term inside the parenthesis is an integer.
Let
be a prime less than
. The exponent of
in the denominator is given by
In the numerator, the terms that are multiples of
are as follows:
Factoring out a
from all these terms gives us
In all, there are
multiples of
in the numerator. Of these
are multiples of
.
Factoring out a
from these terms gives us
.
Continuing in this fashion, we can compute the exponent of
in the numerator as follows:
.
Comparing the exponent of
in the numerator and the denominator , it suffices to show
.
It is enough to show
.
Suppose
divides
. Then
.
Suppose
does not divide
. Then
(i.e.)
where
is a fraction less than
.
Hence,
.
Therefore,
is an integer. In fact, a perfect square.
