Find all pairs (x,y) of positive integers that satisfy this equation:

• x^{2}+y^{2} = 1997(x-y)

Provide valid reasoning as to why there are no further solutions.

I'll start by moving everything to the left side, multiply through by 4, and completing the square. This transforms the equation into

(2x - 1997)^2 + (2y + 1997)^2 = 2*1997^2.

Then integer solutions to the original equation correspond to lattice points laying on the circle m^2 + n^2 = 2*1997^2. These lattice points are easy to find without resorting to computer searches by using the Gaussian integer technique I used in

Imagine the Points.

Factoring over Gaussian integers yields 2*1997^2 = -i*(1+i)^2*(34+29i)^2*(34-29i)^2. There are then 12 distinct factorizations over Gaussian integers; so then we will find a total of 12 lattice points:

(m, n) = (+/-1997, +/-1997), (+/-1657,+/-2287), and (+/-2287,+/-1657).

Now we can take note of the requirement that both x and y are positive. Then for n=2y+1997 we have y=(n-1997)/2>0. Then we must have n>1997. This restriction discards 10 of our 12 lattice points, leaving only (m,n)=(1657,2287) and (-1657,2287). These lattice points correspond to** the pairs (x,y) of positive integers that satisfy the original equation: (x,y)=(1827,145) and (170,145)**.