MHB Can Two Numbers Satisfy These Exponential Conditions?

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The discussion explores finding two distinct numbers, x and y, such that the sum of their squares equals a cube (z^3) and the sum of their cubes equals a square (w^2). The equations x^2 + y^2 = z^3 and x^3 + y^3 = w^2 are analyzed, leading to the simplification where y is expressed in terms of w and z. By setting w as z^2, it is deduced that y equals z, resulting in the pairs (0,0), (0,1), and (1,0) as potential solutions. The conversation emphasizes the importance of transforming real-world applications into mathematical equations, highlighting a key mathematical skill. The exploration confirms at least one valid pair that satisfies the given conditions.
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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?
 
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The cube and the square on the RHS need not be of the same number.
 
greg1313 said:
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
 
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. :)
 
Very impressive reply. This is, in my opinion, the best math skill to master. The ability to transform applications to equations is uniquely important.
 
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