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1997 Crossed Difference Product Poser (Posted on 2023-05-22) Difficulty: 3 of 5
Find all pairs (x,y) of positive integers that satisfy this equation:

•   x2+y2 = 1997(x-y)

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

See The Solution Submitted by K Sengupta    
Rating: 5.0000 (1 votes)

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Solution Analytic Solution Comment 2 of 2 |
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).

  Posted by Brian Smith on 2023-05-22 10:06:34
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