Another proof, by the Swiss mathematician Leonhard Euler, relies on the fundamental theorem of arithmetic: that every integer has a unique prime factorization. If P is the set of all prime numbers, Euler wrote that:
in the result, every product of primes appears exactly once, so by the fundamental theorem of arithmetic, the sum is equal to the sum over all integers.
The sum on the right is the harmonic series, which diverges. So the product on the left must diverge also. Since each term of the product is finite, the number of terms must be infinite, so there is an infinite number of primes.
Read more about this topic: Euclid's Theorem
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