The equation $a^{4n}+b^{4n}+c^{4n}=2d^2$

Problem 3

This is a scheme to generate the solutions which, like your example of $(7,7,12,113)$, have two of $a,b,c$ equal.

Consider the following system of three closely related equations.

E: $2x^4-y^4=z^2$

F: $x^4+8y^4=z^2$

G: $x^4-2y^4=z^2$

A 'base solution' $(x,y,z)$ of E can be used to generate a solution $(z,xy,2x^4+y^4)$ of F.

Each solution $(x,y,z)$ of F can be used to generate a solution $(z,2xy,|x^4-8y^4|)$ of G.

Each solution $(x,y,z)$ of G can be used to generate a further solution $(z,xy,x^4+2y^4)$ of F.

Each solution $(x,y,z)$ of F can be used to generate the solution $(x,x,2y,z)$ of the required equation.

Example starting with the solution $(1,1,1)$ of E.

The scheme generates F$(1,1,3)$, G$(3,2,7)$,F$(7,6,113)$, G$(113,84,7967)$, F$(7967,9492,262621633)$, .....

The required solutions are then $$(1,1,2,3),(7,7,12,113),(7967,7967,18984,262621633),...$$