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

sttscitrans_at_tesco.net
Date: 03/29/05 

Date: 29 Mar 2005 08:41:11 -0800

Nathan wrote:
> Archimedes Plutonium wrote:
> > It is worth repeating, that in the direct proof method of
increasing
> > cardinality a prime factor search must be undertaken. But in the
> > indirect proof method, the moment P!+1 is formed, is the exact
moment
> > that the contradiction arises and that P!+1 is necessarily a new
> > prime. So when one does the Indirect IP and starts to make a search
> > for a "prime factor", well the person has failed and made a invalid
> > proof.
>
> I have 2 (rhetorical) questions:
>
> 1. Why in the world would you think P!+1 must necessarily be a new
> prime? In reality, it scarcely ever is.

"In reality", P!+1 must be divisible by some prime, but in the
"possible world" AP is considering there are only
n primes. In this "possible world", P!+1 is not divisible by any prime
whatsoever as P!+1 = 1 mod pk. Since P!+1 is not equal to 1, P!+1 can't
be an integer. To say that P!+1 is prime or composite or divisible by
some other prime
greater than pn is irrelevant.
AP is, therefore, being "illogical" himself.
GCD(P!+1,p) = 1, for any p,p a prime => P!+1 = 1 => P!=0