SEARCH HOME
 Math Central Quandaries & Queries
 Question from Luke, a student: The "TnS" is a positive integer which is both Triangular number and Square number. For example, 36 is a TnS number since 36=1+2+3+4+5+6+7+8 (Triangular number) and 36=6x6 (Square number). What is the next TnS which is greater than 36?

Hi Luke,

Note that triangular numbers are of the form $\frac{n(n-1)}{2}$ (so actually depending on the parity of $n, \frac{n}{2} \times (n-1)$ or $n \times \frac{(n-1)}{2}).$ The second thing to note is that $n$ and $n-1$ are relatively prime, i.e., they have no common divisors beyond 1. So if a number is a perfect square and a triangular number what does that say about each of the factors $\frac{n}{2}$ and $(n-1)$ (or $n$ and $\frac{n-1}{2}$)? Once you see this you will narrow your search down and find the next one quickly.

Penny

Math Central is supported by the University of Regina and The Pacific Institute for the Mathematical Sciences.