A triangular square number is a number which is both a triangular number and a perfect square. There is an infinity of triangular squares, given by the formula

The problem of finding triangular square numbers reduces to Pell's equation in the following way. Every triangular number is of the form n(n − 1)/2. Therefore we seek integers n, m such that

n(n - 1) / 2 = m².

With a bit of algebra this becomes

(2n - 1)² = 2m² + 1,

and then letting k = 2n − 1, we get the Diophantine equation

k² = 2m² + 1

which is an instance of Pell's equation.

The k^th triangular square N_k is equal to the s^th perfect square and the t^th triangular number, such that

t is given by the formula

.

As k becomes larger, the ratio t/s approaches the square root of two:

External references

• Triangular numbers that are also square (http://www.cut-the-knot.org/do_you_know/triSquare.shtml). From Interactive Mathematics Miscellany and Puzzles.