# Square and Tetrahedral Numbers

## 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