Editing Prime number (section)

=== Analytical proof of Euclid's theorem ===
[[Divergence of the sum of the reciprocals of the primes|Euler's proof that there are infinitely many primes]] considers the sums of [[Multiplicative inverse|reciprocals]] of primes,
: <math>\frac 1 2 + \frac 1 3 + \frac 1 5 + \frac 1 7 + \cdots + \frac 1 p.</math>

Euler showed that, for any arbitrary [[real number]] {{tmath|x}}, there exists a prime {{tmath|p}} for which this sum is greater than {{tmath|x}}.<ref>{{harvnb|Apostol|1976}}, Section 1.6, Theorem 1.13</ref> This shows that there are infinitely many primes, because if there were finitely many primes the sum would reach its maximum value at the biggest prime rather than growing past every&nbsp;{{tmath|x}}.
The growth rate of this sum is described more precisely by [[Mertens' theorems|Mertens' second theorem]].<ref>{{harvnb|Apostol|1976}}, Section 4.8, Theorem 4.12</ref> For comparison, the sum
: <math>\frac 1 {1^2} + \frac 1 {2^2} + \frac 1 {3^2} + \cdots + \frac 1 {n^2}</math>
does not grow to infinity as {{tmath|n}} goes to infinity (see the [[Basel problem]]). In this sense, prime numbers occur more often than squares of natural numbers,
although both sets are infinite.<ref name="mtb-invitation">{{cite book|title=An Invitation to Modern Number Theory|first1=Steven J.|last1=Miller|first2=Ramin|last2=Takloo-Bighash|publisher=Princeton University Press|year=2006|isbn=978-0-691-12060-7|pages=43–44|url=https://books.google.com/books?id=kLz4z8iwKiwC&pg=PA43}}</ref> [[Brun's theorem]] states that the sum of the reciprocals of [[twin prime]]s,
: <math> \left( {\frac{1}{3} + \frac{1}{5}} \right) + \left( {\frac{1}{5} + \frac{1}{7}} \right) + \left( {\frac{1}{{11}} + \frac{1}{{13}}} \right) +  \cdots, </math>
is finite. Because of Brun's theorem, it is not possible to use Euler's method to solve the [[twin prime conjecture]], that there exist infinitely many twin primes.<ref name="mtb-invitation"/>