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Pythagorean Legs (Posted on 2016-10-13) Difficulty: 3 of 5
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.

No Solution Yet Submitted by K Sengupta    
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Solution A solution | Comment 1 of 3

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

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