Let us look at a proof that the number of primes is infinite — a proof that uses only elementary calculus.
Let be the number of primes that are less than or equal to the real number x. (Note: here P is the set of primes). We number the primes in increasing order. Consider the natural logarithm defined as .
Now we compare the area below the graph of with an upper step function. Thus, for we have
where the sum extends over all which have only prime divisors .
Since every such m can be written in a unique way as a product of the form , we see that the last sum is equal to
The inner sum is a geometric series with ratio , hence,
Now, clearly and thus
Everybody knows that is not bounded, so we conclude that is unbounded as well, and so there are infinitely many primes. QED.
Ref: Proofs from THE BOOK (Martin Aigner and Gunter M. Ziegler) (Third Edition).
More about primes later,