Squares equal to Sum of 2 Cubes

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Sequence

The sequence of integers whose square can be expressed as the sum of $2$ coprime cubes begins:

$3, 228, 671, 1261, 6371, 9765, 35 \, 113, 35 \, 928, 40 \, 380, 41 \, 643, 66 \, 599, \ldots$

This sequence is A099426 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).


Proof

\(\displaystyle 3^2\) \(=\) \(\displaystyle 9\)
\(\displaystyle \) \(=\) \(\displaystyle 8 + 1\)
\(\displaystyle \) \(=\) \(\displaystyle 2^3 + 1^3\)


\(\displaystyle 228^2\) \(=\) \(\displaystyle 51 \, 984\)
\(\displaystyle \) \(=\) \(\displaystyle 50 \, 653 + 1331\)
\(\displaystyle \) \(=\) \(\displaystyle 37^3 + 11^3\)


\(\displaystyle 671^2\) \(=\) \(\displaystyle 450 \, 241\)
\(\displaystyle \) \(=\) \(\displaystyle 274 \, 625 + 175 \, 616\)
\(\displaystyle \) \(=\) \(\displaystyle 65^3 + 56^3\)



Sources