[HM] Re: First incommensurables, Theaetetus

Prof. Lueneburg (luene@mathematik.uni-kl.de)
Tue, 22 Dec 1998 09:35:29 +0100 (MEZ)

>
> >There exist, of course, purely geometric proofs of the incomensurability of
> >the diagonal with the side, but I do not dispose of exact references at the
> >moment, nor dates thereof.
> >AJFO.
> >At 10:09 21-12-1998 -0500, you wrote:
> >>On Mon, 21 Dec 1998, Fernando Q. Gouvea wrote:
> >>>
> >>> It certainly seems natural to reconstruct the proof(s) as you have... but
> >>> aren't there other possibilities? I think it is Grattan-Guinness who
> >>> suggested that the "odd numbers" that are equal to "even" in Aristotle
> >>> could also be the *number of prime divisors* of a^2=2b^2. I have no idea
> >>> how probable it is that the Greeks could have reasoned in this way, but it
> >>> does give a proof too.
> >>
> >> But one that depends on the theory of unique factorization into
> >>primes, which it's far from clear was available at the time. (Euclid's
> >>statement is a weaker one.)
> >>
> >> John Conway
> >>
> >>
>
>
If one analyzes the proof given by Euclid, one sees that one needs the following
properties of the multiplication of natural numbers. 1) Associativity.
2) 2a = a2 for all a. 3) given n, then n = 2^t.o with o odd. 4) Given a divisor
d of n = 2^t.o, then d = 2^s.p with s <= t and p a divisor of o. (Divisor
meaning right divisor) This is, of course, a precursor of unique factorization,
but much easier to prove. The analysis [I added 2). One does not need the full
commutativity] is in

Oskar Becker, Die Lehre vom Geraden und Ungeraden im neunten Buch der
euklidischen Elemente. Quellen und Studien zur Geschichte der Mathematik,
Astronomie und Physik. Abteilung B: Studien. Band 3, S. 533--553, Berlin 1935.

Reprinted in

Oskar Becker, Zur Geschichte der griechischen Mathematik. Unveraenderte
Teilauflage der ersten Auflage 1965. Darmstadt 1980

These properties suffice to construct all even perfect numbers and to show that
one has really got them all (modulo Mersenne primes, of course).

Heinz Lueneburg