## A Little Challenge

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• If I had to give my #1 reason for my interest in prime numbers, I d have to say that it goes back to my first reading Euclid s proof of their infinity. How
Message 1 of 1 , Dec 2, 2001
If I had to give my #1 reason for my interest in prime numbers, I'd have
to say that it goes back to my first reading Euclid's proof of their
infinity. How simple and concise a proof of such magnitude that it took
so many, many years before it was discovered. Elegance is the word
describing such an effort. The fact that there are so many as yet
unsolved conjectures involving the primes is impetus to keep one's
interest in the topic alive.

But now to a slightly different topic: I still vividly recall a theorem
involving the real numbers which also has an elegant and fairly short
(but quite difficult to come up with quickly) proof. I posted this a
while ago, but got no replies. In any case, it would be an interesting
challenge to see if you can come up with it. If anyone wants the
solution, just email me at mweissm2@... ---- Marty Weissman

Statement of problem:

Let S represent a set of real numbers (called a "POSITIVE CLASS") with
the following
properties: (1) For every real number x, either x belongs to S, -x
belongs to S, or x = 0, and all
three are mutually exclusive. (2) If x and y belong to S, then so does x

+ y and xy. Clearly, the
set of positive real numbers form a "POSITIVE CLASS". The problem is to

prove that this is,
in fact, the only set of real numbers which do form a "POSITIVE CLASS".
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