Question:
Pythagorean Triangles
Please help me with the following problem:
a) 3^2 + 4^2 = 5^2
20^2 + 21^2 = 29^2
119^2 + 120^2 = 169^2
To find another such relation, show that if a^2 + (a+1)^2 = c^2, then
(3a+2c+1)^2 + (3a+2c+2)^2 = (4a+3c+2)^2.
(b) If a^2 + (a+1)^2 = c^2, let u=c-a-1 and v=(2a+1-c)/2. Show that v is an integer and that u(u+1)/2 = v^2. This shows that there are infinitely many square triangular numbers.
Please show as much detail as possible.