Tetrahedral and Triangular Numbers

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Theorem

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

$1, 10, 120, 1540, 7140$


Proof

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


\(\displaystyle 10\) \(=\) \(\displaystyle \dfrac {3 \paren {3 + 1} \paren {3 + 2} } 6\) Closed Form for Tetrahedral Numbers
\(\displaystyle \) \(=\) \(\displaystyle \dfrac {4 \times \paren {4 + 1} } 2\) Closed Form for Triangular Numbers


\(\displaystyle 120\) \(=\) \(\displaystyle \dfrac {8 \paren {8 + 1} \paren {8 + 2} } 6\) Closed Form for Tetrahedral Numbers
\(\displaystyle \) \(=\) \(\displaystyle \dfrac {15 \times \paren {15 + 1} } 2\) Closed Form for Triangular Numbers


\(\displaystyle 1540\) \(=\) \(\displaystyle \dfrac {20 \paren {20 + 1} \paren {20 + 2} } 6\) Closed Form for Tetrahedral Numbers
\(\displaystyle \) \(=\) \(\displaystyle \dfrac {55 \times \paren {55 + 1} } 2\) Closed Form for Triangular Numbers


\(\displaystyle 7140\) \(=\) \(\displaystyle \dfrac {34 \paren {34 + 1} \paren {34 + 2} } 6\) Closed Form for Tetrahedral Numbers
\(\displaystyle \) \(=\) \(\displaystyle \dfrac {119 \times \paren {119 + 1} } 2\) Closed Form for Triangular Numbers



Sources