# Triangular Numbers which are Product of 3 Consecutive Integers

## Theorem

The $6$ triangular numbers which can be expressed as the product of $3$ consecutive integers are:

$6, 120, 210, 990, 185 \, 836, 258 \, 474 \, 216$

## Proof

 $\displaystyle T_3$ $=$ $\displaystyle \frac {3 \left({3 + 1}\right)} 2$ Closed Form for Triangular Numbers $\displaystyle$ $=$ $\displaystyle 6$ $\displaystyle$ $=$ $\displaystyle 1 \times 2 \times 3$

 $\displaystyle T_{15}$ $=$ $\displaystyle \frac {15 \left({15 + 1}\right)} 2$ Closed Form for Triangular Numbers $\displaystyle$ $=$ $\displaystyle 120$ $\displaystyle$ $=$ $\displaystyle 4 \times 5 \times 6$

 $\displaystyle T_{20}$ $=$ $\displaystyle \frac {20 \left({20 + 1}\right)} 2$ Closed Form for Triangular Numbers $\displaystyle$ $=$ $\displaystyle 210$ $\displaystyle$ $=$ $\displaystyle 5 \times 6 \times 7$

 $\displaystyle T_{44}$ $=$ $\displaystyle \frac {44 \left({44 + 1}\right)} 2$ Closed Form for Triangular Numbers $\displaystyle$ $=$ $\displaystyle 990$ $\displaystyle$ $=$ $\displaystyle 9 \times 10 \times 11$

 $\displaystyle T_{608}$ $=$ $\displaystyle \frac {608 \left({608 + 1}\right)} 2$ Closed Form for Triangular Numbers $\displaystyle$ $=$ $\displaystyle 185 \, 136$ $\displaystyle$ $=$ $\displaystyle 56 \times 57 \times 58$

 $\displaystyle T_{22 \, 736}$ $=$ $\displaystyle \frac {22 \, 736 \left({22 \, 736 + 1}\right)} 2$ Closed Form for Triangular Numbers $\displaystyle$ $=$ $\displaystyle 258 \, 474 \, 216$ $\displaystyle$ $=$ $\displaystyle 2^3 \times 3 \times 7^2 \times 11 \times 13 \times 29 \times 53$ $\displaystyle$ $=$ $\displaystyle \left({2^2 \times 3 \times 53}\right) \times \left({7^2 \times 13}\right) \times \left({2 \times 11 \times 29}\right)$ $\displaystyle$ $=$ $\displaystyle 636 \times 637 \times 638$