Heptagonal Pyramidal Numbers which are Square
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Theorem
The sequence of heptagonal pyramidal numbers which also have the property of being square begins:
- $0, 1, 196, 99 \, 225$
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Proof
\(\ds \) | \(\) | \(\ds \dfrac {0 \paren {0 + 1} \paren {5 \times 0 - 2} } 6\) | Closed Form for Heptagonal Pyramidal Numbers | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {0 \times 1 \times \paren {- 2} } 6\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 0\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 0^2\) | Definition of Square Number |
\(\ds \) | \(\) | \(\ds \dfrac {1 \paren {1 + 1} \paren {5 \times 1 - 2} } 6\) | Closed Form for Heptagonal Pyramidal Numbers | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {1 \times 2 \times 3} 6\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 1\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 1^2\) | Definition of Square Number |
\(\ds \) | \(\) | \(\ds \dfrac {6 \paren {6 + 1} \paren{5 \times 6 - 2} } 6\) | Closed Form for Heptagonal Pyramidal Numbers | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {6 \times 7 \times 28} 6\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 7 \times \paren {2^2 \times 7}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \paren {2 \times 7}^2\) | Definition of Square Number | |||||||||||
\(\ds \) | \(=\) | \(\ds 196\) |
\(\ds \) | \(\) | \(\ds \dfrac {49 \paren {49 + 1} \paren{5 \times 49 - 2} } 6\) | Closed Form for Heptagonal Pyramidal Numbers | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {49 \times 50 \times 243} 6\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 7^2 \times \paren {5^2 \times 9^2}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \paren {5 \times 7 \times 9}^2\) | Definition of Square Number | |||||||||||
\(\ds \) | \(=\) | \(\ds 99 \, 225\) |
$\blacksquare$
Sources
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $196$