Concrete mathematics : a foundation for computer science

4.3 PRIME EXAMPLES 111
These formulas, which hold only in the limit as n or x + 03, can be
replaced by more exact estimates. For example, Rosser and Schoenfeld [253]
have established the handy bounds
lnx-i < * < lnx-t, for x 3 67; (4.19)
n(lnn+lnlnn-3) < P, < n(lnn+lnlnn-t), forn320. (4.20)
If we look at a “random” integer n, the chances of its being prime are
about one in Inn. For example, if we look at numbers near 1016, we’ll have to
examine about 16 In 10 x 36.8 of them before finding a prime. (It turns out
that there are exactly 10 primes between 1016 - 370 and 1016 - 1.) Yet the
distribution of primes has many irregularities. For example, all the numbers
between PI PZ P, + 2 and P1 PJ . . . P, + P,+l - 1 inclusive are composite.
Many examples of “twin primes” p and p + 2 are known (5 and 7, 11 and 13,
17and19,29and31, . . . . 9999999999999641 and 9999999999999643, . . . ), yet
nobody knows whether or not there are infinitely many pairs of twin primes.
(See Hardy and Wright [150, $1.4 and 52.81.)
One simple way to calculate all X(X) primes 6 x is to form the so-called
sieve of Eratosthenes: First write down all integers from 2 through x. Next
circle 2, marking it prime, and cross out all other multiples of 2. Then repeatedly
circle the smallest uncircled, uncrossed number and cross out its other
multiples. When everything has been circled or crossed out, the circled numbers
are the primes. For example when x = 10 we write down 2 through 10,
circle 2, then cross out its multiples 4, 6, 8, and 10. Next 3 is the smallest
uncircled, uncrossed number, so we circle it and cross out 6 and 9. Now
5 is smallest, so we circle it and cross out 10. Finally we circle 7. The circled
numbers are 2, 3, 5, and 7; so these are the X( 10) = 4 primes not exceeding 10.
“Je me sers de la
z”;$ Zg$;/f 4.4 FACTORIAL FACTORS
produif de nombres
dkroissans depuis
n jusqu9 l’unitk,
saioir-n(n - 1)
(n - 2). 3.2.1.
L’emploi continue/
de l’analyse combinatoire
que je fais
dans /a plupart de
mes dCmonstrations,
a rendu cette notation
indispensa b/e. ”
- Ch. Kramp (186]
Now let’s take a look at the factorization of some interesting highly
composite numbers, the factorials:
n! = 1.2...:n = fib integer n 3 0. (4.21)
k=l
According to our convention for an empty product, this defines O! to be 1.
Thus n! = (n - 1 )! n for every positive integer n. This is the number of
permutations of n distinct objects. That is, it’s the number of ways to arrange
n things in a row: There are n choices for the first thing; for each choice of
first thing, there are n - 1 choices for the second; for each of these n(n - 1)
choices, there are n - 2 for the third; and so on, giving n(n - 1) (n - 2) . . . (1)