Find all values of a positive integer constant A such that x²+Ax and Ax+A describes two non hypotenuse legs of a Pythagorean triangle for every positive integer value of x.

A=2 is a solution:

Let A = 2
Then (x^2+2x)^2+(2x+2)^2 = y^2
Let y=((x+1)^2+1)
Then (x^2+2x)^2+(2x+2)^2 = ((x+1)^2+1)^2, true for all x.

A=2 is the only solution:

Now let x = 3, say, then 25A^2 - y^2 + 54A + 81 = 0 has only the solution A=2 in the positive integers.

Edited on October 14, 2016, 8:53 am

Posted by broll on 2016-10-14 01:22:20