Determine the total number of pairs (x,y) of positive integers that satisfy this equation:
√(x)+ √(y) = √(1988)
Rearrange the equation to isolate x:
sqrt[x] = sqrt[1988]-sqrt[y]
Then square each side:
x = 1988+y - 2*sqrt[1988y]
x and y are integers, then we must have 1988y is a perfect square. 1988=2^2*497. 497 is square free, thus for 1988y to be a perfect square then y is 497 times a perfect square.
An identical argument can be used to show that x is also 497 times a perfect square.
Then let x=497v^2 and y=497w^2. Then the equation becomes
sqrt[497v^2] + sqrt[497w^2] = sqrt[1988]
This reduces to v + w = 2. Over the positive integers this has one solution, (v,w)=(1,1). Then back substituting we get (x,y)=(497,497) is the only pair of positive integers that satisfies the given equation.
Analytic Solution
