[FOM] Only one proof

[Rob Arthan]

On 10 Sep 2009, at 01:38, Vaughan Pratt wrote:

> Tim Chow (and others) wrote:
>> There's a proof, sometimes called the "Hasse-Lindemann-Zermelo"  
>> proof,
>> that proceeds directly by induction without reference to GCDs.
>
> That's the only proof I know, viz:
>
> Let the least counterexample be pr = qs with p<q prime and r,s  
> nonempty
> strings of primes. By leastness p does not appear in s, and cannot
> divide q-p, whence p(r-s) = (q-p)s yields a smaller counterexample.
>
> I find it hard to believe Fermat or Euler would have considered this a
> "new proof," and I would be shocked if Gauss had.  It's simply the
> distillation to its essence of the only way it's ever been argued.
....

See Hardy & Wright's "Introduction to the Theory of Numbers": section  
2.11 and notes thereon for the references to Hasse, Lindemann and  
Zermelo's and section 2.10 for the quite different proof of the  
fundamental theorem of algebra using GCDs.

Regards,

Rob.