Smallest Integer which is Sum of 2 Cubes in 4 Ways

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Theorem

The smallest positive integer which can be expressed as the sum of $2$ cubes in $4$ different ways is:

\(\ds 42 \, 549 \, 416\) \(=\) \(\ds 348^3 + 74^3\)
\(\ds \) \(=\) \(\ds 282^3 + 272^3\)
\(\ds \) \(=\) \(\ds \left({-2662}\right)^3 + 2664^3\)
\(\ds \) \(=\) \(\ds \left({-475}\right)^3 + 531^3\)


Proof



Sources