Who was the mathematician who proved there are infinite prime numbers?

Answer 1

The first mathematician to do this may have been Euclid, but we can't be sure he was the first.

The following proof is due to Euler.

Let zeta(s) = sum of reciprocals of the s-th powers of the natural

numbers.

Because every number has a unique prime factorization, we have

zeta(s) = product over all primes p of (1+1/p^s+1/p^(2s) + ...)

which in turn can be written

zeta(s) = product over all prime p of 1/(1-1/p^s)

(an application of the formula for an infinite geometric series)

Note that the limit of zeta(1+e) as e tends to 0 is infinity.

This can be seen by making a comparison of the series of reciprocals

of the s-th powers of the naturals with an integral of 1/x^s.

If there were only finitely many primes, then this limit should

converge to the finite product over all p of 1/(1-1/p^s), a

contradiction.

Dirichlet generalized this idea to obtain his theorem that any

arithmetic series with base relatively prime to the common difference

contains infinitely many primes.

For another proof, you can show that the sums of the reciprocals of

the primes diverges...see Hardy and Littlewood's book on number

theory.

-Doctor Ceeks, The Math Forum.