Here is the question:
I have posted a link there to this topic so the OP can see my work.X^2 + 1 = 2y^2 Diophantine equation?
I've been thinking about what whole numbers x and y satisfy:
x^2 + 1 = 2y^2
I know that (1, 1) works, as do (7, 5) and (41, 29), but I want a formula that gives me the next one, or any given one. What's the pattern?
The question is equivalent to asking which triangular numbers are twice other ones (the first three are (0, 0), (3, 2) (3rd is 6, 2nd is 3), and (20, 14)), and which right triangles with integer legs have one leg 1 unit longer than the other (first three triplets are (0, 1, 1), (3, 4, 5), (20, 21, 29)).
I'm not sure how to solve it. Which numbers x and y satisfy it? Is there an equation to find, say, the 20th x and y? Are there an infinite number of pairs that satisfy it, or just a few? Please help.