# Thread: Integer solutions to a^2+b^2=c^3

1. ## Integer solutions to a^2+b^2=c^3

a^2+b^2=c^3, a,b,c are positive integers, and b>=a.

Just by inspection, a=b=c=2 is one set of solution.

Also, a=26, b=18, c=10; a=1358, b=594, c=130; are two other sets of solutions.

If you multiply a and b by n^3 and c by n^2 for any positive integers n, you can easily get related sets of solutions.

So my the question is:
- Do other solutions exists?
- Are there any solutions, such that a,b,c are pairwise coprime, exists?

2. One solution set is:

$(a,b,c) = (|x^3 - 3xy^2|,|3x^2y-y^3|, x^2+y^2)$ (and removing those where b<a)

Nor is it completely unmotivated. Consider the Gaussian integers:

$a^2+b^2 = c^3 \Rightarrow (a+ib)(a-ib) = c^3$

Suppose $a+ib = (x+iy)^3 = x^3 - 3xy^2 + i(3x^2y-y^3)$