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Prove that any positive integral power of $2$ is representable in the form $7x^2+y^2$. where $x,y$ are integers.

Base Case:
$2^3=8=7+1=7(1)^2+1^2$
$2^4=16=7+9=7(1)^2+3^2$

Induction:
For odd powers,
Let $2^p=7a^2+b^2$[$p$ is odd]
$\rightarrow 2^{p+2}=7(2a)^2+(2b)^2$

For even powers,
Let $2^p=7a^2+b^2$[$p$ is even]
$\rightarrow 2^{p+2}=7(2a)^2+(2b)^2$

Proved by induction.

For even powers, take $x=0$ and for odd powers take $x=y$. We're done!
Note that through this solution we can also find a representation directly.