Integer as Sum of 2 Cubes in 3 Ways

From ProofWiki
Jump to navigation Jump to search

Theorem

$4104$ is the smallest natural number which can be expressed as the sum of $2$ cubes in $3$ different ways:

\(\ds 4104\) \(=\) \(\ds 16^3 + 2^3\)
\(\ds \) \(=\) \(\ds 15^3 + 9^3\)
\(\ds \) \(=\) \(\ds \paren {-12}^3 + 18^3\)


Proof

We have:

\(\ds 16^3 + 2^3\) \(=\) \(\ds 4096 + 8\)
\(\ds \) \(=\) \(\ds 4104\)


\(\ds 15^3 + 9^3\) \(=\) \(\ds 3375 + 729\)
\(\ds \) \(=\) \(\ds 4104\)


\(\ds \paren {-12}^3 + 18^3\) \(=\) \(\ds -1728 + 5832\)
\(\ds \) \(=\) \(\ds 4104\)




Sources