The distribution of prime numbers is highly irregular but consideration of large blocks of prime numbers we can iron out the irregularities to some extent. Although there are infinitely many primes, as we go along the list of prime numbers we find they become more widely spaced. If we define the function
is defined as 'the number of primes p satisfying
then we can draw up the following table.
|
|
|
|
|
10 |
4 |
0.93 |
|
|
25 |
1.15 |
|
|
168 |
1.16 |
|
|
1229 |
1.13 |
|
|
9592 |
1.1 |
|
|
78498 |
1.08 |
|
|
664579 |
1.07 |
|
|
5761455 |
1.06 |
|
|
50847534 |
1.05 |
|
|
455052511 |
1.05 |
By considering the behaviour of
for large
Gauss and Legendre proposed that this quotient tended to 1. This is the Prime Number Theorem.
Proofs of the prime number theorem are not simple, involving tricky limits or products of the type
fact it took almost 100 years for the theorem to be proved because several intermediate problems needed to be solved first.