Math Olympiad Question | Nice Algebra Equation Solving | How To Solve for a+b !!

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[@RexxSchneider]

That formula only works when c = b + 1. Check out b + 1 = (n^2 - 1)/2 + 1 = (n^2 + 1)/2 = c. Note that a has to be an odd integer because the difference between consecutive squares is odd.
If you're going to guess that that particular formula works, then you might as well guess b = c-1 = 612 and you find that a = √(613^2 - 612^2) = 35 immediately.
You can make lots of formulae for Pythagorean triples by setting c=b+2 or c=b+3, etc. For example pick n to be even then a = n; b = (n^2 - 4)/4; and c = (n^2 + 4)/4 = b+2.
If you want c=b+k, then a^2 = c^2-b^2 = 2bk+k^2. So you get a Pythagorean triple whenever you can pick k such that b = (a^2-k^2)/2k is an integer.

[@danielduranloosli]

Wouldn't you need to state all pythagorean triplets have that form (2n; n^2-1; n^2+1) in order to be sure those are all the possible solutions?

[@BabySarkar-tc1vt]

Only for the positive integers solution,, I think this is the formula