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Proof: Infinitely Many Primes


Date: 6/24/96 at 23:5:0
From: Anonymous
Subject: Proof: Infinitely Many Primes

I need another proof that there are an infinite number of primes other 
than Euclid's proof by contradiction.

What is the Dirichlet proof?


Date: 9/4/96 at 14:15:12
From: Doctor Ceeks
Subject: Re: Proof: Infinitely Many Primes

Hi,

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
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Associated Topics:
College Number Theory

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