Pell's Equation
The problem of finding square triangular numbers reduces to Pell's equation in the following way. Every triangular number is of the form t(t + 1)/2. Therefore we seek integers t, s such that
With a bit of algebra this becomes
and then letting x = 2t + 1 and y = 2s, we get the Diophantine equation
which is an instance of Pell's equation. This particular equation is solved by the Pell numbers Pk as
and therefore all solutions are given by
There are many identities about the Pell numbers, and these translate into identities about the square triangular numbers.
Read more about this topic: Square Triangular Number
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