# A Pythagorean quadruple-like equation: $(2a^2)^2+(ab)^2+2^2=c^2$

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.

• It might help to tell us where you found this equation. What makes you think that a general solution exists? – 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 $$(a,b,c)=(n,n^3,n^4+2)\quad\forall n\in\mathbb{N}$$
• Clearly not for example $$(a,b,c)=(3, 13, 43)$$I don't believe every possible solution is part of a single generalisation. – Peter Foreman Oct 1 '19 at 23:36