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

34

THEORY OF NUMBERS

From our hypothesis it follows that the number of integers
less than m and divisible by at least one of the primes
pi, p2,..., pi i
or
Z-m + - -.., (A)
pl pIp2 plp2p3
where the summation in each case runs over all numbers of
the type indicated, the subscripts of the p's being equal to or
less than i.
Let us consider the integers less than m and having the
factor pi+i but not having any of the factors pi, p2, *... pi.
Their number is.^- I m _ m (B)
p+l1 pi+l pi Plp2 plp2p3 v
where the summation signs have the same significance as before.
For the number in question is evidently m/p+ 1 minus the
number of integers not greater than m/pi+l and divisible by
at least one of the primes pi, p2,. ~., pi.
If we add (A) and (B) we have the number of integers less
than m and divisible by one at least of the numbers pi, P2,..., pi+. Hence the number of integers less than m and
prime to pi, P2,..., pI+i is
m m m m
m-E- + - -E +,
Pi Plp2 plp2P3
where now in the summations the subscripts run from i to
i+i. This number is clearly equal to
I I/ I
i Pl ( 2 pi+ 1
From this result, as we have seen above, our theorem follows
at once by induction.