Triangular Numbers which are Sum of Two Cubes
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Theorem
The sequence of triangular numbers which are the sum of $2$ cubes begins:
- $28, 91, 351, 2926, 8001, 46971, 58653, 93528, 97461, \dots$
This sequence is A113958 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).
Proof
Can be demonstrated by brute force.
For example:
\(\ds 28\) | \(=\) | \(\ds 1 + 27\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 1^3 + 3^3\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {7 \paren {7 + 1} } 2\) |
\(\ds 91\) | \(=\) | \(\ds 27 + 64\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 3^3 + 4^3\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {13 \paren {13 + 1} } 2\) |
\(\ds 351\) | \(=\) | \(\ds 125 + 216\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 5^3 + 6^3\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {26 \paren {26 + 1} } 2\) |
\(\ds 2976\) | \(=\) | \(\ds 729 + 2197\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 5^3 + 6^3\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {76 \paren {76 + 1} } 2\) |
\(\ds 8001\) | \(=\) | \(\ds 1 + 8000\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 1^3 + 20^3\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {126 \paren {126 + 1} } 2\) |
\(\ds 46 \, 971\) | \(=\) | \(\ds 4096 + 42 \, 875\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 16^3 + 35^3\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {306 \paren {306 + 1} } 2\) |
\(\ds 58 \, 653\) | \(=\) | \(\ds 8000 + 50 \, 653\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 20^3 + 37^3\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {342 \paren {343 + 1} } 2\) |
\(\ds 93 \, 528\) | \(=\) | \(\ds 42 \, 875 + 50 \, 653\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 35^3 + 37^3\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {432 \paren {433 + 1} } 2\) |
\(\ds 97 \, 461\) | \(=\) | \(\ds 125 + 97 \, 336\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 5^3 + 46^3\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {441 \paren {442 + 1} } 2\) |
$\blacksquare$
Sources
- 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $28$