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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.)

 Aug 24, 2023
edited by tomtom  Aug 24, 2023
 #1
avatar+6 
+1

Rearranging the equation, we have:
x^2 - 2xy + y^2 = 17

Factoring the left side of the equation, we get:
(x - y)^2 = 17

To find the lattice points, we need to find integer solutions for (x, y) that satisfy the equation.

Since 17 is a prime number, its only factors are 1 and 17. Therefore, the possible values for (x - y) are ±1 and ±17. Vampire Survivors

For (x - y) = ±1:
If (x - y) = 1, then we have the solution (x, y) = (1, 0).
If (x - y) = -1, then we have the solution (x, y) = (-1, 0).

For (x - y) = ±17:
If (x - y) = 17, then we have the solution (x, y) = (17, 0).
If (x - y) = -17, then we have the solution (x, y) = (-17, 0).

Therefore, there are four lattice points that lie on the graph of the given equation: (1, 0), (-1, 0), (17, 0), and (-17, 0).

 Aug 25, 2023
 #2
avatar+128409 
0

Probably an infinite  number

 

I found these   (3, 4) , (11, 26)  , (39, 94)  and  (-3, -4) , (-11, -26), (-39 -94) 

 

cool cool cool  

 Aug 25, 2023

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