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
and therefore,
.
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,
Nalin Pithwa
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