# Tetrahedral and Triangular Numbers

## 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