Concrete mathematics : a foundation for computer science

4.4 FACTORIAL FACTORS 115
We can use this observation to get another proof that there are infinitely
many primes. For if there were only the k primes 2, 3, . . . , Pk, then we’d
have n! < (2”)k = 2nk for all n > 1, since each prime can contribute at most
a factor of 2” - 1. But we can easily contradict the inequality n! < 2”k by
choosing n large enough, say n = 22k. Then
contradicting the inequality n! > nn/2 that we derived in (4.22). There are
infinitely many primes, still.
We can even beef up this argument to get a crude bound on n(n), the
number of primes not exceeding n. Every such prime contributes a factor of
less than 2” to n!; so, as before,
n! < 2nn(n).
If we replace n! here by Stirling’s approximation (4.23), which is a lower
bound, and take logarithms, we get
hence
nrr(n) > nlg(n/e) + i lg(27rn) ;
This lower bound is quite weak, compared with the actual value z(n) -
n/inn, because logn is much smaller than n/logn when n is large. But we
didn’t have to work very hard to get it, and a bound is a bound.
4.5 RELATIVE PRIMALITY
When gcd(m, n) = 1, the integers m and n have no prime factors in
common and we say that they’re relatively prime.
This concept is so important in practice, we ought to have a special
notation for it; but alas, number theorists haven’t come up with a very good
one yet. Therefore we cry: HEAR us, 0 MATHEMATICIANS OF THE WORLD!
Like perpendicular
LETUS NOTWAITANYLONGER! W E CAN MAKEMANYFORMULAS CLEARER
BY DEFINING A NEW NOTATION NOW! LET us AGREE TO WRITE ‘m I n’,
lines don ‘t have
a common direction,
perpendicular
numbers don’t have
AND TO SAY U, IS PRIME TO Tl.;
In other words, let us declare that
IF m AND n ARE RELATIVELY PRIME.
common factors. ml-n w m,n are integers and gcd(m,n) = 1, (4.26)