Sum of 2 Squares in 2 Distinct Ways which is also Sum of Cubes
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
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $65$