Tetrahedral and Triangular Numbers
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Theorem
The only positive integers which are simultaneously tetrahedral and triangular are:
- $1, 10, 120, 1540, 7140$
Proof
\(\ds 1\) | \(=\) | \(\ds \dfrac {1 \paren {1 + 1} \paren {1 + 2} } 6\) | Closed Form for Tetrahedral Numbers | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {1 \times \paren {1 + 1} } 2\) | Closed Form for Triangular Numbers |
\(\ds 10\) | \(=\) | \(\ds \dfrac {3 \paren {3 + 1} \paren {3 + 2} } 6\) | Closed Form for Tetrahedral Numbers | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {4 \times \paren {4 + 1} } 2\) | Closed Form for Triangular Numbers |
\(\ds 120\) | \(=\) | \(\ds \dfrac {8 \paren {8 + 1} \paren {8 + 2} } 6\) | Closed Form for Tetrahedral Numbers | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {15 \times \paren {15 + 1} } 2\) | Closed Form for Triangular Numbers |
\(\ds 1540\) | \(=\) | \(\ds \dfrac {20 \paren {20 + 1} \paren {20 + 2} } 6\) | Closed Form for Tetrahedral Numbers | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {55 \times \paren {55 + 1} } 2\) | Closed Form for Triangular Numbers |
\(\ds 7140\) | \(=\) | \(\ds \dfrac {34 \paren {34 + 1} \paren {34 + 2} } 6\) | Closed Form for Tetrahedral Numbers | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {119 \times \paren {119 + 1} } 2\) | Closed Form for Triangular Numbers |
This theorem requires a proof. In particular: It remains to be shown that these are the only such instances. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{ProofWanted}} from the code.If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page. |
Sources
- 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $1540$
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $1540$