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
\(\ds 3^2\) | \(=\) | \(\ds 9\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 8 + 1\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 2^3 + 1^3\) |
\(\ds 228^2\) | \(=\) | \(\ds 51 \, 984\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 50 \, 653 + 1331\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 37^3 + 11^3\) |
\(\ds 671^2\) | \(=\) | \(\ds 450 \, 241\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 274 \, 625 + 175 \, 616\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 65^3 + 56^3\) |
\(\ds 1261^2\) | \(=\) | \(\ds 1 \, 590 \, 121\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 185 \, 193 + 1 \, 404 \, 928\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 57^3 + 112^3\) |
\(\ds 6371^2\) | \(=\) | \(\ds 40 \, 589 \, 641\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 10 \, 218 \, 313 + 30 \, 371 \, 328\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 217^3 + 312^3\) |
\(\ds 9765^2\) | \(=\) | \(\ds 95 \, 355 \, 225\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 14 \, 172 \, 488 + 81 \, 182 \, 737\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 242^3 + 433^3\) |
\(\ds 35 \, 113^2\) | \(=\) | \(\ds 1 \, 232 \, 922 \, 769\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 28 \, 372 \, 625 + 1 \, 204 \, 550 \, 144\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 305^3 + 1064^3\) |
\(\ds 35 \, 928^2\) | \(=\) | \(\ds 1 \, 290 \, 821 \, 184\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 616 \, 295 \, 051 + 674 \, 526 \, 133\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 851^3 + 877^3\) |
\(\ds 40 \, 380^2\) | \(=\) | \(\ds 1 \, 630 \, 544 \, 400\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 12 \, 167 + 1 \, 630 \, 532 \, 233\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 23^3 + 1177^3\) |
\(\ds 41 \, 643^2\) | \(=\) | \(\ds 1 \, 734 \, 139 \, 449\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 1 \, 815 \, 848 + 1 \, 732 \, 323 \, 601\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 122^3 + 1201^3\) |
\(\ds 66 \, 599^2\) | \(=\) | \(\ds 4 \, 435 \, 426 \, 801\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 207 \, 474 \, 688 + 4 \, 227 \, 952 \, 113\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 592^3 + 1617^3\) |
$\blacksquare$
Sources
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $51,984$