Square and Tetrahedral Numbers

From ProofWiki
Jump to navigation Jump to search

Theorem

The only positive integers which are simultaneously tetrahedral and square are:

$1, 4, 19 \, 600$


Proof

\(\displaystyle 1\) \(=\) \(\displaystyle \dfrac {1 \paren {1 + 1} \paren {1 + 2} } 6\) Closed Form for Tetrahedral Numbers
\(\displaystyle \) \(=\) \(\displaystyle 1^2\) Definition of Square Number


\(\displaystyle 4\) \(=\) \(\displaystyle \dfrac {2 \paren {2 + 1} \paren {2 + 2} } 6\) Closed Form for Tetrahedral Numbers
\(\displaystyle \) \(=\) \(\displaystyle 2^2\) Definition of Square Number


\(\displaystyle 19 \, 600\) \(=\) \(\displaystyle \dfrac {48 \paren {48 + 1} \paren {48 + 2} } 6\) Closed Form for Tetrahedral Numbers
\(\displaystyle \) \(=\) \(\displaystyle 140^2\) Definition of Square Number



Sources