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WOLFRAM|DEMONSTRATIONS PROJECT

Enumerating Pythagorean Triangles

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n
392
rational
14
47
triangle {2013,1316,2405}
There is a one-to-one correspondence between positive rational numbers
q
less than 1 and points with positive rational coordinates
(x,y)
on the unit circle. This correspondence is achieved by joining the point
(-1,0)
with
(0,q)
and extending the line to intersect the unit circle at
(x,y)
as shown in this Demonstration. As any integral solution of the equation
2
a
+
2
b
=
2
c
corresponding to a Pythagorean triangle can be put in the form
2
(a/c)
+
2
(b/c)
=1
, we can associate Pythagorean triangles with points with positive rational coordinates on the unit circle. This Demonstration shows the
th
n
rational number and its associated
th
n
Pythagorean triangle. By varying
n
, can you find the only Pythagorean triangle with a side equal to 2009 that exists in the given range? Alas, the first rational with a part equal to 2009 is 30/2009 and it occurs at
n=154876
, too far out of our range
n<1000
.
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