Square Pyramidal and Triangular Numbers

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Theorem

The only positive integers which are simultaneously square pyramidal and triangular are:

$1, 55, 91, 208 \, 335$

This sequence is A039596 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).


Proof

\(\displaystyle 1\) \(=\) \(\displaystyle \dfrac {1 \paren {1 + 1} \paren {2 \times 1 + 1} } 6\) Closed Form for Square Pyramidal Numbers
\(\displaystyle \) \(=\) \(\displaystyle \dfrac {1 \times \paren {1 + 1} } 2\) Closed Form for Triangular Numbers


\(\displaystyle 55\) \(=\) \(\displaystyle \dfrac {5 \paren {5 + 1} \paren {2 \times 5 + 1} } 6\) Closed Form for Square Pyramidal Numbers
\(\displaystyle \) \(=\) \(\displaystyle \dfrac {10 \times \paren {10 + 1} } 2\) Closed Form for Triangular Numbers


\(\displaystyle 91\) \(=\) \(\displaystyle \dfrac {6 \paren {6 + 1} \paren {2 \times 6 + 1} } 6\) Closed Form for Square Pyramidal Numbers
\(\displaystyle \) \(=\) \(\displaystyle \dfrac {13 \times \paren {13 + 1} } 2\) Closed Form for Triangular Numbers


\(\displaystyle 208 \, 335\) \(=\) \(\displaystyle \dfrac {85 \paren {85 + 1} \paren {2 \times 85 + 1} } 6\) Closed Form for Square Pyramidal Numbers
\(\displaystyle \) \(=\) \(\displaystyle \dfrac {645 \times \paren {645 + 1} } 2\) Closed Form for Triangular Numbers



Sources