Pentagonal and Hexagonal Numbers
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Theorem
The sequence of positive integers which are simultaneously pentagonal and hexagonal begins:
- $1, 40 \, 755, 1 \, 533 \, 776 \, 805, 57 \, 722 \, 156 \, 241 \, 751, \ldots$
This sequence is A046180 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).
Proof
| \(\ds 1\) | \(=\) | \(\ds \dfrac {1 \paren {3 \times 1 - 1} } 2\) | Closed Form for Pentagonal Numbers | |||||||||||
| \(\ds \) | \(=\) | \(\ds 1 \paren {2 \times 1 - 1}\) | Closed Form for Hexagonal Numbers |
| \(\ds 40 \, 755\) | \(=\) | \(\ds \dfrac {165 \paren {3 \times 165 - 1} } 2\) | Closed Form for Pentagonal Numbers | |||||||||||
| \(\ds \) | \(=\) | \(\ds 143 \paren {2 \times 143 - 1}\) | Closed Form for Hexagonal 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): $40,755$
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $40,755$