Assume that the sum of the reciprocals of the primes converges:
Define <math>p_i</math> as the ith prime number. We have:
There exists a positive integer i such that:
Define N(x) as the number of positive integers n not exceeding x and not divisible by a prime other than the first i ones. Let us write this n as <math>km^2</math> with k squarefree (which can be done with any integer). Since there are only i primes which could divide k, there are at most <math>2^i</math> choices for k. Together with the fact that there are at most <math>\sqrt{x}</math> possible values for m, this gives us:
The number of positive integers not exceeding x and divisible by a prime other than the first i ones is equal to x  N(x).
Since the number of integers not exceeding x and divisible by p is at most x/p, we get:
or:
But this is impossible for all x larger than (or equal to) <math> 2^{2i+2} </math>.
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