So i have just started to study number theory and i was asked this question.
Now i tried to search online and i found out pell equation can used to solve this question. Now in an online video i saw that $$x^2-2y^2=1$$ is the equation and you take the solution point such as the first solution $x_1=3$ and $y_1=2$ and you input that into $T_n=S_n$ and solving that you will get your first triangular square number.
My question is how do you go about finding the solutions to pell equation without doing trial by error.
Please correct me if i am wrong or got some misinformation
The first two numbers that are both squares and triangles are $1$ and $36$.
- Find the next one and, if possible, the one after that.
- Can you figure out an efficient way to find triangular-square numbers?
- Do you think that there are infinitely many?