The theory of numbers, by Robert D. Carmichael ...

ELEMENTARY PROPERTIES OF INTEGERS

25

Applying the same process to the H-function in the second
member and remembering 'relation (i) it is easy to see that
we have
Hin>! [>I P2* P}
\n~\, \n 1. ~ f f n it
+p+.p2J (.3... p2
Continuing the process we have finally
=PI I + [P
the series on the right containing evidently only a finite number of terms different from zero. Thus we have the theorem:
I. The index of the highest power of a prime p contained
in n. is
In j+n V, |+
The theorem just obtained may be written in a different
form, more convenient for certain of its applications. Let
n be expressed in the scale of p in the form
n=aoph+-alph-l+.. +ah-ip+a7z,
where
ao0o, o<ai<p, i=o, i, 2,..., h.
Then evidently
li=paoph+alph~+... +ah-2p+ah-1,
P -
F ] aoph-2 + a ph -- 3+ +ah-2,