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Find the number of lattice points that lie on the graph of $x^2-y^2=17-2xy$. (A lattice point is a point in the coordinate plane where both coordinates are integers.)

 Dec 11, 2023
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Adding 2xy to both sides of the equation x2−y2=17−2xy gives us (x+y)2=17, so 17​=x+y. Since x+y must be an integer because x and y are integers, we have 1≤x+y≤4. Therefore, (x+y)2=1 or (x+y)2=4.

 

If (x+y)2=1, then x+y=±1. If x+y=1, then the possible values of x and y are:

(0,1)

(1,0)

(−1,2)

(2,−1)

 

If x+y=−1, then the possible values of x and y are:

(−1,−2)

(−2,−1)

(0,−1)

(1,0)

 

If (x+y)2=4, then x+y=±2. If x+y=2, then the possible values of x and y are:

(0,2)

(1,1)

(2,0)

(−1,3)

(−2,4)

 

If x+y=−2, then the possible values of x and y are:

(−1,−3)

(−2,−4)

(0,−2)

(1,−1)

(2,0)

 

Therefore, there are 19​ lattice points that lie on the graph of x2−y2=17−2xy.

 Dec 11, 2023

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