Concrete mathematics : a foundation for computer science

110 NUMBER THEORY
But 2P - 1 isn’t always prime when p is prime; 2” - 1 = 2047 = 23.89 is the
smallest such nonprime. (Mersenne knew this.)
Factoring and primality testing of large numbers are hot topics nowadays.
A summary of what was known up to 1981 appears in Section 4.5.4 of [174],
and many new results continue to be discovered. Pages 391-394 of that book
explain a special way to test Mersenne numbers for primality.
For most of the last two hundred years, the largest known prime has
been a Mersenne prime, although only 31 Mersenne primes are known. Many
people are trying to find larger ones, but it’s getting tough. So those really
interested in fame (if not fortune) and a spot in The Guinness Book of World
Records might instead try numbers of the form 2nk + 1, for small values of k
like 3 or 5. These numbers can be tested for primality almost as quickly as
Mersenne numbers can; exercise 4.5.4-27 of [174] gives the details.
We haven’t fully answered our original question about how many primes
there are. There are infinitely many, but some infinite sets are “denser” than
others. For instance, among the positive integers there are infinitely many
even numbers and infinitely many perfect squares, yet in several important
senses there are more even numbers than perfect squares. One such sense
looks at the size of the nth value. The nth even integer is 2n and the nth
perfect square is n’; since 2n is much less than n2 for large n, the nth even
integer occurs much sooner than the nth perfect square, so we can say there
are many more even integers than perfect squares. A similar sense looks at
the number of values not exceeding x. There are 1x/2] such even integers and
L&j perfect squares; since x/2 is much larger than fi for large x, again we
can say there are many more even integers.
What can we say about the primes in these two senses? It turns out that
the nth prime, P,, is about n times the natural log of n:
pll N nlnn.
(The symbol ‘N’ can be read “is asymptotic to”; it means that the limit of
the ratio PJnlnn is 1 as n goes to infinity.) Similarly, for the number of
primes n(x) not exceeding x we have what’s known as the prime number
theorem:
Proving these two facts is beyond the scope of this book, although we can
show easily that each of them implies the other. In Chapter 9 we will discuss
the rates at which functions approach infinity, and we’ll see that the function
nlnn, our approximation to P,, lies between 2n and n2 asymptotically.
Hence there are fewer primes than even integers, but there are more primes
than perfect squares.
Weird. I thought
there were the same
number of even
integers as perfect
squares, since
there’s a one-to-one
correspondence
between them.