Solving for $k$: $k^2=x+y$ and $k^3=x^2+y^2$

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SUMMARY

The discussion focuses on solving the equations $k^2 = x + y$ and $k^3 = x^2 + y^2$ for non-negative integers $k$, $x$, and $y$. It concludes that the only solutions occur when $k = 0$ and $k = 1$. For $k = 0$, both $x$ and $y$ must be zero, while for $k = 1$, $x$ and $y$ can be either both 0 or both 1. No other integer values for $k$ yield valid integer pairs $(x, y)$.

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Find all non-negative integers $k$ such that there are integers $x$ and $y$ with the property

$k^2=x+y$ and $k^3=x^2+y^2$
 
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Hint:

Compare $2(x^2+y^2)$ and $(x+y)^2$.
 

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