Heptagonal Pyramidal Numbers which are Square

From ProofWiki
Jump to navigation Jump to search

Theorem

The sequence of heptagonal pyramidal numbers which also have the property of being square begins:

$0, 1, 196, 99 \, 225$




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