are all integers divisible by . This follows from the fact that is expressible in terms of those symmetrical functions which
consist of the sums of products of the numbers ; and
these expressions have integral values.
(4) Let denote the integer
,
which may be written in the form
.
In virtue of what has been established in (3) as to the values of
we see that is not a multiple of .
"We examine the form to which the equation
is reduced by multiplying all the terms by .
We have
r=np+p-I
r=p-l
Q r+i O r+2
"*" T "*" 7 ~ / ^T +
The modulus of the sum of the series
does not exceed
,
and this is less than ; hence we have
,
where is some number whose modulus is between 0 and 1.
The modulus of is less than ,