I am looking for a general solution to

$$(2a^2)^2+(ab)^2+4=c^2$$ where $a,b,c, \in\mathbb{Z^+}$

In fact, it is possible to find the values ​​that work. But, I am looking for a general solution. At least, I want to know if there are infinitely many solutions to this equation.

My effort is only research and brute-force. I found that this equation looks like Pythagorean quadruple.But, I have no idea what kind of method can be applied to the equation above. And I don't know if this equation can be solved by elementary number theory.I don't know anything about the Method. So, it's pointless to talk about my steps.

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    $\begingroup$ It might help to tell us where you found this equation. What makes you think that a general solution exists? $\endgroup$ – pokep Oct 1 '19 at 22:59

Taking $(a,b)=(n,n^3)$ for any $n\in\mathbb{N}$, we have $$(2n^2)^2+(n\cdot n^3)^2+4=4n^4+n^8+4=(n^4+2)^2$$ So we have infinitely many solutions given by $$(a,b,c)=(n,n^3,n^4+2)\quad\forall n\in\mathbb{N}$$

  • $\begingroup$ (+1) Impressive work. This answer provides the first part of my question. Am I right? $\endgroup$ – Learner Oct 1 '19 at 23:26
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    $\begingroup$ I mean Your work provide an answer a question of "Are there infinitely many solutions?" . We say Yes. But , Are these solutions contain all possible solution? $\endgroup$ – Learner Oct 1 '19 at 23:33
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    $\begingroup$ Clearly not for example $$(a,b,c)=(3, 13, 43)$$I don't believe every possible solution is part of a single generalisation. $\endgroup$ – Peter Foreman Oct 1 '19 at 23:36
  • $\begingroup$ I understood Thank you very much for your answer. $\endgroup$ – Learner Oct 1 '19 at 23:38

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