Re: G.H.Hardy gave an invalid proof of Infinitude of Primes in "A Mathematician's Apology"

From: Arthur Fischer (arthurfischer_at_sym.ca)
Date: 03/29/05 

Date: Tue, 29 Mar 2005 10:44:35 -0500

Archimedes Plutonium wrote:

> --- first posted by me in the early to mid 1990s to the Internet ---
>
> But my business here today is to analyze Hardy's attempt to prove
> Euclid's Infinitude of Primes. As in the title of this post, Hardy
> failed, but the bright light is that one can see his pretty poetry. I
> quote from his book.
> --- quoting A MATHEMATICIAN'S APOLOGY, Godfrey H. Hardy, 1940,
> pages 92-94 ---
>
> [snip of quotation]
>
> Let us suppose that it does, and that 2,3,5,..., P is the complete
> series (so that P is the largest prime); and let us, on this
> hypothesis, consider the number Q defined by the formula Q =
> (2x3x5x..xP) + 1. It is plain that Q is not divisible by any of
> 2,3,5,...,P; for it leaves the remainder 1 when divided by any one of
> these numbers. But, if not itself prime, it is divisible by some prime,
> and therefore there is a prime (which may be Q itself) greater than any
> of them. This contradicts our hypothesis, that there is no prime
> greater than P; and therefore this hypothesis is false.
>
> [snip of quotation]
>
> --- end quoting A MATHEMATICIAN'S APOLOGY, Godfrey H. Hardy, 1940,
> pages 92-94 ---
>
> Hardy's flaw is that Q is NECESSARILY prime and once Q is formed it is
> the
> end of the proof since it forms the contradiction that simultaneously
> not prime and prime; or, that Q is a larger prime than P.

Suppose that we thought, wrongly, that the only prime numbers were
2,3,5,7,11,13,and 17 (which is the starting point of Hardy's proof ---
we think that we have some finite list that enumerates all primes; the
fact that I have chosen a ridiculously short initial segment of the
sequence of primes is, of course, immaterial). Then simple calculation
of Hardy's method gives
Q = 2x3x5x7x11x13x17 + 1 = 510511 = 19(26869)

Are you claiming that there are natural numbers which are both prime and
composite?

__
Arthur