Re: Irrationality and the Fundamental Theorem of Arithmetic

In article
<2825276.1152533007084.JavaMail.jakarta@xxxxxxxxxxxxxxxxxxxxxx>,
"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.

There's a geometric version of the same proof that John Conway
came up with a few years ago. Being a geometric proof, it's best
understood with the aid of diagrams, and such can be found on
pages 2 and 3 of http://www.nd.edu/~hahn/pdf%20files/Ch1-Addit.pdf
but I'll try to give it in words:

Suppose n^2 = 2 m^2, m, n integers.
Draw a square of side n, and put a square of side m in
the lower left corner, and another in the upper right corner.
This creates three squares along the upper left to lower right
diagonal; one in the middle, where the two m-squares overlap,
and one in each corner, that the m-squares miss. All three
of these squares have integer sides, and the two identical
corner ones must add up to the middle one. This gives a smaller
solution to x^2 = 2 y^2, and you get a proof by descent.

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

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