I would like to find a parametric solution for the following diophantine equation:

$-4 (1-a_1^2)(1-a_2^2) + (1+a_3^2 -a_1^2 -a_2^2)^2 = a_4^2$

Does such a solution exist? How does one go about solving such questions systematically?

  • $\begingroup$ Could you write this equation in this form? $q^2=(z^2+d^2-x^2-y^2)^2-4(d^2-x^2)(d^2-y^2)$ And look for a solution in integers. Because this equation has a solution. $\endgroup$ – individ May 30 '14 at 17:19
  • $\begingroup$ Yes indeed, this amounts to a rescaling of my variables. Where can I find a solution to that equation? $\endgroup$ – Johannes May 30 '14 at 17:23
  • $\begingroup$ Nowhere. It is necessary to solve this equation. $\endgroup$ – individ May 30 '14 at 17:24
  • $\begingroup$ Ok, I see. Are there somewhat systematic methods for solving equations like this? Perhaps it is better to start with a simpler version of this. E.g., the following eq. is also relevant for me, $(1 - b2^2) (1 - b3^2) - 4 (1 - b1^2) = 0$. It has one variable less. $\endgroup$ – Johannes Jun 2 '14 at 13:29
  • $\begingroup$ For such equations the more unknown so easily solved. It is better not to reduce the number of unknowns. And what is the equation you need? $\endgroup$ – individ Jun 2 '14 at 14:00



Has the solution:









$p,s,a,t,k$ - Any integers.


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