Can Two Numbers Satisfy These Exponential Conditions?

[mathdad]

Find two different numbers such that the sum of their squares shall equal a cube, and the sum of their cubes equal a square.

Set up:

x^2 + y^2 = z^3
x^3 + y^3 = z^2

Is this correct?

 

[Greg]

The cube and the square on the RHS need not be of the same number.

 

[mathdad]

The cube and the square on the RHS need not be of the same number.

x^2 + y^2 = z^3
x^3 + y^3 = w^2

 

[MarkFL]

Let's let $x=0$ so that we have:

$$y^2=z^3$$

$$y^3=w^2$$

Dividing the latter by the former, we have:

$$y=\frac{w^2}{z^3}$$

Suppose we let $w=z^2$...

$$y=z$$

This implies:

$$y^3-y^2=y^2(y-1)=0$$

Because of the cyclical symmetry, this yields:

$$(x,y)\in\{(0,0),(0,1),(1,0)\}$$

There may or may not be more pairs that work, but we have found at least one pair satisfying the problem. :)

 

[mathdad]

Very impressive reply. This is, in my opinion, the best math skill to master. The ability to transform applications to equations is uniquely important.