Tonico, I blush to admit that it was I myself who inadvertantly put this
bee
in Archie's bonnet. I remarked here at sci.math that there are several
differences between Euclid's proof of IX.20
http://aleph0.clarku.edu/~djoyce/java/elements/bookIX/propIX20.html
and the modern version, although the basic ideas are same and the proof is
still attributed to Euclid. In particular,
-- Euclid dares not speak of "infinitely many" primes (or any other actual
infinite) as we do.
-- Euclid works on what we would call an arbitrary finite non-empty family
of primes. He does not argue by contradiction and hypothesize that they
are
/all/ the primes, and he does not assume that they are distinct.
Anyhow, here's a proof of IX.20 discovered by Filip Saidak in 2005 AD!
Define inductively a sequence (x(n), y(n)) of ordered pairs by
x(1) = 2
y(1) = 3
and for n >= 2 :
x(n+1) = x(n)y(n)
y(n+1) = x(n)y(n) + 1
By induction, x and y stay positive. Also x(n) and y(n) are relatively
prime
by definition. So y(n) has a prime factor which x(n) lacks, and that
factor,
along with all the factors of x(n), appears in x(n+1). So by induction
x(n)
has at least n distinct prime factors, for arbitrary n.
LH
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