[FOM] Only one proof

Rob Arthan rda at lemma-one.com
Fri Sep 11 10:11:57 EDT 2009

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.



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