Square and Tetrahedral Numbers
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Theorem
The only positive integers which are simultaneously tetrahedral and square are:
- $1, 4, 19 \, 600$
Proof
\(\ds 1\) | \(=\) | \(\ds \dfrac {1 \paren {1 + 1} \paren {1 + 2} } 6\) | Closed Form for Tetrahedral Numbers | |||||||||||
\(\ds \) | \(=\) | \(\ds 1^2\) | Definition of Square Number |
\(\ds 4\) | \(=\) | \(\ds \dfrac {2 \paren {2 + 1} \paren {2 + 2} } 6\) | Closed Form for Tetrahedral Numbers | |||||||||||
\(\ds \) | \(=\) | \(\ds 2^2\) | Definition of Square Number |
\(\ds 19 \, 600\) | \(=\) | \(\ds \dfrac {48 \paren {48 + 1} \paren {48 + 2} } 6\) | Closed Form for Tetrahedral Numbers | |||||||||||
\(\ds \) | \(=\) | \(\ds 140^2\) | Definition of Square Number |
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): $19,600$
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $19,600$