Concrete mathematics : a foundation for computer science

Or probably more,
by the time you
read this.
4.3 PRIME EXAMPLES 109
Thus e, has about twice as many decimal digits as e,-1 . Exercise 37 proves
that there’s a constant E z 1.264 such that
And exercise 60 provides a similar formula that gives nothing but primes:
Pn = lp3"J ,
(4.17)
for some constant P. But equations like (4.17) and (4.18) cannot really be
considered to be in closed form, because the constants E and P are computed
from the numbers e, and p,, in a sort of sneaky way. No independent relation
is known (or likely) that would connect them with other constants of
mathematical interest.
Indeed, nobody knows any useful formula that gives arbitrarily large
primes but only primes. Computer scientists at Chevron Geosciences did,
however, strike mathematical oil in 1984. Using a program developed by
David Slowinski, they discovered the largest prime known at that time,
2216091 -1
while testing a new Cray X-MP supercomputer. It’s easy to compute this
number in a few milliseconds on a personal computer, because modern computers
work in binary notation and this number is simply (11 . . .1)2. All
216 091 of its bits are ‘1'. But it’s much harder to prove that this number
is prime. In fact, just about any computation with it takes a lot of time,
because it’s so large. For example, even a sophisticated algorithm requires
several minutes just to convert 22’609’ - 1 to radix 10 on a PC. When printed
out, its 65,050 decimal digits require 65 cents U.S. postage to mail first class.
Incidentally, 22’609’ - 1 is the number of moves necessary to solve the
Tower of Hanoi problem when there are 216,091 disks. Numbers of the form
2p - 1
(where p is prime, as always in this chapter) are called Mersenne numbers,
after Father Marin Mersenne who investigated some of their properties in the
seventeenth century. The Mersenne primes known to date occur for p = 2, 3,
5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607, 1279, 2203, 2281, 3217, 4253,
4423, 9689,9941, 11213,19937,21701, 23209,44497, 86243,110503, 132049,
and 216091.
The number 2” - 1 can’t possibly be prime if n is composite, because
2k” - 1 has 2”’ - 1 as a factor:
2km - 1 = (2" - l)(2mckp') +2"'+2) +...+ 1).