structural:expansion, information:highlighting
Analytical proof of Euclid's theorem

Euler's proof that there are infinitely many primes considers the sums of reciprocals of primes,

Euler showed that, for any arbitrary real number x {\displaystyle x}, there exists a prime p {\displaystyle p} for which this sum is bigger than x {\displaystyle x}. 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 x {\displaystyle x}.The growth rate of this sum is described more precisely by Mertens' second theorem. For comparison, the sum

does not grow to infinity as n {\displaystyle 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. Brun's theorem states that the sum of the reciprocals of twin primes,

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.