Integer as Sum of 2 Cubes in 3 Ways
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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:
\(\ds 16^3 + 2^3\) | \(=\) | \(\ds 4096 + 8\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 4104\) |
\(\ds 15^3 + 9^3\) | \(=\) | \(\ds 3375 + 729\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 4104\) |
\(\ds \paren {-12}^3 + 18^3\) | \(=\) | \(\ds -1728 + 5832\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 4104\) |
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Sources
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $4104$