Let (x, y) = (a+1/2, b+1/2)
Then, [x]^2+ [y]^2 =2016, gives:
a^2+b^2+a+b =2016
=> (2a+1)^2+ (2b+1)^2 = 8066
=> (2a+1, 2b+1) = (85,29}, (85, -29), (-85, 29), (-85, -29), (71,55), (71, -55), (55,71), (55, -71), (-55, 71), (-55, -71), (29,85), (29, -85), (-29, 85), ( -85), (55
-73),
(a, b) = (-42,14), (-42, -15), (35, -28), (27, 35)
,(27, -36), (-28, 35), (14, -43), (55, -73
(27, 35), (27, -36), (14, 42), (14, -43)
Now, we know that: (a,b) = (x+1/2, y+1/2)
Therefore, we have:
(x,y) = (29/2, 85/2), (55/2, 71/2), (-29/2, 85/2),
(-55/2, 71/2), (-71/2, 55/2), (-85/2, 29/2), (-85/2, -29/2), (-71/2, -55/2), as all possible solutions to the given problem.
For an alternate methodology, refer to the solution
submitted by Charlie here and here.
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