Sum of 2 Squares in 2 Distinct Ways which is also Sum of Cubes/Sequence

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Theorem

The sequence of positive integers which are both the sum of $2$ square numbers in two distinct ways and also the sum of $2$ cube numbers begins:

$65, 250, \ldots$


Proof

\(\ds 65\) \(=\) \(\ds 4^3 + 1^3\)
\(\ds \) \(=\) \(\ds 8^2 + 1^2\)
\(\ds \) \(=\) \(\ds 7^2 + 4^2\)


\(\ds 250\) \(=\) \(\ds 5^3 + 5^3\)
\(\ds \) \(=\) \(\ds 15^2 + 5^2\)
\(\ds \) \(=\) \(\ds 13^2 + 9^2\)

$\blacksquare$


Historical Note

This result is attributed by David Wells in his Curious and Interesting Numbers, 2nd ed. of $1997$ to a source going by the name of Thayer, but it has not been possible to find out any further information.


Sources