Enumerating Pythagorean Triangles

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

.