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


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Sources

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