At first glance, this paper seems to be spot on! Carsten Eisner. ON RATIONAL APPROXIMATIONS BY PYTHAGOREAN NUMBERS. May, 2003. I'll go over it more carefully. Thanks for the link, Dan! At 03:36 PM 8/14/2013, Dan Asimov wrote:
P.P.S. This paper may shed some light on both the continued-fraction Pythagorean-triple question and the uncontinued-fraction Pythagorean-triple question:
< http://www.fq.math.ca/Scanned/41-2/elsner.pdf >.
--Dan
On 2013-08-14, at 3:21 PM, Dan Asimov wrote:
P.S. As may have been clear, I was addressing only Henry's second sentence quoted below.
But the existence of arbitrarily good approximations says nothing about how one might go about finding them.
QUESTION: Given an irrational real number c, (say, given all the decimal digits of c), how would we explicitly find a sequence {p_n / q_n} of rationals approaching c in the limit as n -> oo, *such that* for each n there exists an integer r_n with
(p_n)^2 + (q_n)^2 = (r_n)^2
???
--Dan
On 2013-08-14, at 11:04 AM, Dan Asimov wrote:
This is a very interesting question. We know that the rational points on the unit circle are all of the form (a/c, b/c) where (a,b,c) is a Pythagorean triple. And we know these are dense in the unit circle, so this approximation must be possible. (In fact these rational points form an interesting subgroup of the unit circle group SO(2).)
--Dan
On 2013-08-14, at 10:23 AM, Henry Baker wrote:
Suppose I'm trying to compute some rational approximation to a real number x, but with some additional constraint.
In particular, if m/n approximates x, I'd like sqrt(m^2+n^2) to be integral.