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I am working on Project Euler 390.

The question is about triangles, and finding the area of a triangle with sides $\sqrt{a^2+1}, \sqrt{b^2+1}$ and $\sqrt{a^2+b^2}$, with $a, b \in \mathbb{Z}$. I have narrowed the problem down to solving the equation

$$x^2 \cdot y^2 + x^2 + y^2 = (2\cdot c)^2 \text{ with } x, y, c \in \mathbb{Z}^+$$

This is not a problem for $c \le 10^6$ (brute force), but I have to calculate up to $10^{10}$. I would like to know how to solve these kind of equations, without any brute force attack. I have searched for a few days on Google, but the general solutions to the Diophantine equations I found were never appliable to my problem.

Any suggestions are welcome (even the name of this kind of equation), although I would appreciate not being told the answer to the problem.

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    Oops, you're right. I'll delete my comment.2012-08-15
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    We do have $(2c)^2+1=(x^2+1)(y^2+1)$ which indicates that both factors on the RHS are odd, and x,y are therefore even.2012-08-15
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    @MarkBennet Thanks for the comment, I found that out already. I still require 25 minutes of computing power to calculate all the $(x, y)$ for $c \le 10^6$. $10^{10}$ is just not possible brute force.2012-08-15
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    It seems to me that this curve (for a fixed $c$) is closely related to [Edwards' curves](http://www.ams.org/journals/bull/2007-44-03/S0273-0979-07-01153-6/home.html). May be with a twist (Daniel Bernstein and Tanja Lange have studied the twisted curves)? Anyway, they are birationally equivalent to elliptic curves meaning that parametrizing the rational points may be a tall order.2012-08-15
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    I don't know how much use it is, but another form is $(x+y)^2 + (xy - 1)^2 = 4c^2 + 1$.2012-08-15

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