- #1
geoffrey159
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Homework Statement
Solve in ##\mathbb{Z}^2## the equation ## x^2 -y^2-x+3y = 30 ##
Homework Equations
The Attempt at a Solution
Hello, can you tell me if this is correct please ?
The equation is equivalent to ## (x-y+1) (x+y-2) = 28 ##.
I call ## u = x-y+1 ## and ## v = x+y-2 ##
We have that ## u | 28 ##, ## v | 28 ##, and ##uv = 28##
So, ##u,v \in \text{Div}(28)=\{\pm 1, \pm 2, \pm 4, \pm 7, \pm 28 \} ## and share their sign
Finally, we must have ##(u,v) \in## { (-1,-28) , (1,28), (-2,-14), (2,14), (-4,-7) ,(4,7), (-7,-4), (7,4), (-14,-2), (14,2), (-28,-1),(28,1) }
So ## (x,y) = ( \frac{u+v+1}{2}, \frac{v-u+3}{2} ) \in ## { (-14,-12),(15,15),(-5,0),(6,3),(-5,3), (6,0), (-14,15), (15,-12) }