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

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Theorem

The smallest positive integer which is both the sum of $2$ square numbers in two distinct ways and also the sum of $2$ cube numbers is $65$:

\(\ds 65\) \(=\) \(\, \ds 16 + 49 \, \) \(\, \ds = \, \) \(\ds 4^2 + 7^2\)
\(\ds \) \(=\) \(\, \ds 1 + 64 \, \) \(\, \ds = \, \) \(\ds 1^2 + 8^2\)
\(\ds \) \(\) \(\, \ds = \, \) \(\ds 1^3 + 4^3\)


Sequence

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

From Sum of 2 Squares in 2 Distinct Ways, the smallest $2$ positive integer which are the sum of $2$ square numbers in two distinct ways are $50$ and $65$.

But $50$ cannot be expressed as the sum of $2$ cube numbers:

\(\ds 50 - 1^3\) \(=\) \(\ds 49\) which is not cubic
\(\ds 50 - 2^3\) \(=\) \(\ds 42\) which is not cubic
\(\ds 50 - 3^3\) \(=\) \(\ds 23\) which is not cubic
\(\ds 50 - 4^3\) \(=\) \(\ds -14\) and we have fallen off the end

Hence $65$ is that smallest number.

$\blacksquare$


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