In the third century BC, Euclid proved the existence of infinitely many prime numbers. In the 18th century, Leonhard Euler proved a stronger statement: the sum of the reciprocals of all prime numbers diverges to infinity. A proof by contradiction follows.
Assume that the sum of the reciprocals of the primes converges:
Define as the ith prime number. We have:
Since the number of integers not exceeding x and divisible by p is at most x/p, we get:
Q.E.D
Here is another proof that actually gives an estimate for the sum; in particular, it shows that the sum grows at least as large as ln ln n. The proof is an adaptation of the product expansion idea of Euler. In the following, a sum or product taken over p always represents a sum or product taken over a specified set of primes.
The proof rests upon the following facts:
- Every positive integer n can be expressed as the product of a square-free integer and a square. This gives the inequality
The product corresponds to the square-free part of n and the sum corresponds to the square part of n. (See fundamental theorem of arithmetic.)
- The inequality
which can be obtained by considering approximating rectangles in the integral definition of ln n. (See natural logarithm.)
- The inequality 1 + x < exp(x), which holds for all x > 0. (See exponential.)
- The identity
Actually, the exact sum is not necessary; we just need to know that the sum converges, and this can be shown using the p-test for series. (See series.)
Combining all these facts, we see that
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