Concrete mathematics : a foundation for computer science

112 NUMBER THEORY
arrangements in all. Here are the first few values of the factorial function.
n 01234 5 6 7 8 9 10
n! 1 1 2 6 24 120 720 5040 40320 362880 3628800
It’s useful to know a few factorial facts, like the first six or so values, and the
fact that lo! is about 34 million plus change; another interesting fact is that
the number of digits in n! exceeds n when n > 25.
We can prove that n! is plenty big by using something like Gauss’s trick
of Chapter 1:
n!’ = (1 .2...:n)(n... :2.1) = fik(n+l-k).
We have n 6 k(n + 1 - k) 6 $ (n + 1 )2, since the quadratic polynomial
k(n+l -k) = a(r~+l)~- (k- $(n+ 1))2 has its smallest value at k = 1 and
its largest value at k = i (n + 1). Therefore
that is,
k=l k=l
n n/2 6 n! <
(n+ l)n
2n .
k=l
(4.22)
This relation tells us that the factorial function grows exponentially!!
To approximate n! more accurately for large n we can use Stirling’s
formula, which we will derive in Chapter 9:
n! N &Gi(:)n. (4.23)
And a still more precise approximation tells us the asymptotic relative error:
Stirling’s formula undershoots n! by a factor of about 1 /( 12n). Even for fairly
small n this more precise estimate is pretty good. For example, Stirling’s
approximation (4.23) gives a value near 3598696 when n = 10, and this is
about 0.83% x l/l20 too small. Good stuff, asymptotics.
But let’s get back to primes. We’d like to determine, for any given
prime p, the largest power of p that divides n!; that is, we want the exponent
of p in n!‘s unique factorization. We denote this number by ep (n!), and we
start our investigations with the small case p = 2 and n = 10. Since lo! is the
product of ten numbers, e:2( lo!) can be found by summing the powers-of-2