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
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $250$