Unusual number triads

Question

Let's have the following equations.
$1388^2+803^2=137^3$
$236^2+115^2=41^3$
$666^2+413^2=85^3$
As you can verify by visual inspection the numbers of each triad have (GCD)=1.
Question 1) How can we obtain more such triads?
Question 2) What is the required method to find such triads?

Bubbler · Accepted Answer

No previous answers addressed the GCD=1 part, so here it is.
First of all, note that

Then use happystar's method to get the general formula for $A^2 + B^2 = C^3$:

So we get

Then the requirement is

The LHS simplifies into

Therefore,

happystar · Answer

The following method works for $41^3$

The same method works for $N^3$ provided

Unusual number triads

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