Re: Irrationality and the Fundamental Theorem of Arithmetic

In article <gerry-62AB1D.11154012072006@xxxxxxxxxxxxxxxxxx>,
Gerry Myerson <gerry@xxxxxxxxxxxxxxxxxxxxxxxxx> wrote:

In article <y8zzmfg1fed.fsf@xxxxxxxxxxxxxxxxxxxx>,
Bill Dubuque <wgd@xxxxxxxxxxxxxxxxxxxx> wrote:

Gerry Myerson <gerry@xxxxxxxxxxxxxxxxxxxxxxxxx> wrote:
"J. B. Kennedy" <john.b.kennedy@xxxxxxxxxxxxxxxx> wrote:

But I then came across a simpler, seemingly more basic proof
ascribed to Niven (which doesn't invoke the FTAr):

If r is rational then
there is a least integer b such that br is an integer
but (br-b) is less than b (since r is btwn 1 and 2)
and then (br-b)r also is an integer,
which contradicts the assumption that b was the least
and so r is irrational.

According to Jorn Steuding, Diophantine Analysis, page 15,
this idea for proving the irrationality of sqrt 2 is due
to Estermann.

A reference is Math Gazette 59 (1975), page 110.

That proof is much, much older than 1975.
I'm shocked any number theorist could believe it so new.
Are you sure the reference is not to a different proof?

The London Mathematical Society published a long obituary
for Estermann. It's on the web, and one page of it,
http://www.numbertheory.org/obituaries/LMS/estermann/page11.html
is relevant here. It says, "In retirement, Estermann discovered
[1975] an elegant proof of Pythagoras's theorem which is actually
simpler than the original and is sufficiently short to be
included verbatim. [note - from what follows, it's evident
that they don't mean the theorem about the square of the hypotenuse,
they mean the irrationality of the square root of two]

They then give Estermann's proof, which is essentially
the one ascribed to Niven up near the top of this post.

I wouldn't be a bit surprised to learn that Niven (or someone)
got there before Estermann, but I don't think anyone in this
thread has cited a reference predating Estermann.

I've had a bit of a look for the proof.

In a thread on sci.math.research in September, 1998, Jim Propp
attributed the proof to Niven, without any references. Michael
Hardy claimed that the proof was actually older than the odd/even
and unique factorization proofs, but I think he was talking
about a geometric proof. The Well-Ordering proof may be nothing
more than the algebraic formulation of that geometric proof, but
I don't accept that the geometric proof *is* the well-ordering
proof.

You can find the Propp and Hardy posts under the Subject header,
The Book.

I didn't find the proof in Niven's book, Numbers: Rational and
Irrational, in the New Mathematical Library series, nor in his
book, Irrational Numbers, in the Carus Mathematical Monographs
series. I searched Math Reviews for anything that had both
Niven and irrational in it, and found nothing that looked like
a reference to this proof.

David Bloom published (a version of) the proof in Mathematics
Magazine (A one-sentence proof that sqrt2 is irrational, Math
Mag 68 (1995) 286). He writes that it is an algebraic version
of a geometric argument in a math history book by Eves, and it
was presented by Niven at a lecture in 1985.

In short, I still haven't found any reference to Niven or any-
one else preceding Estermann's paper in 1975.

--
Gerry Myerson (gerry@xxxxxxxxxxxxxxx) (i -> u for email)
.

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