Math CentralQuandaries & 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.


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