94 6. Integers, Divisors, and Primes
must be at least two, except for 2 and 3. Two primes whose difference is 2
are calledtwin primes.Thus (3,5), (5,7), (11,13), (17,19) are twin primes.
Looking at the table of the primes up to 500, we find many twin primes;
extensive computation shows that there are twin primes with hundreds of
digits. However, it is not known whether there are infinitely many twin
primes! (Almost certainly there are, but no proof of this fact has been
found, in spite of the efforts of many mathematicians for over 2000 years!)
Another way of turning Theorem 6.4.2 around is to ask, how large can
these gaps be, relative to where they are on the number line? Could it
happen that there is no prime at all with, say, 100 digits? This is again a
very difficult question, but here we do know the answer. (No, this does not
happen.)
0
5
10
15
20
25
20 40 60 80 100
FIGURE 6.2. The graph ofπ(n) from 1 to 100.One of the most important questions about primes is, how many primes
are there up to a given numbern? We denote the number of primes up to
nbyπ(n). Figure 6.2 illustrates the graph of this function in the range 1 to
100, and Figure 6.3 shows the range 1 to 2000. We can see that the function
grows reasonably smoothly, and that its slope decreases slowly. An exact
formula forπ(n) is certainly impossible to obtain. Around 1900, a powerful
result called thePrime Number Theoremwas proved by Hadamard and de
la Vall ́ee Poussin.