Otro
Enumerating Pythagorean triangles
Autor
Rangel-Mondragon, Jaime
Resumen
Ensino Médio::Matemática 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 a² + b² = c² corresponding to a Pythagorean triangle can be put in the form (a/c)² + (b/c)²=1, we can associate Pythagorean triangles with points with positive rational coordinates on the unit circle. This Demonstration shows the n^th rational number and its associated n^th 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