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Equation with several variables

  1. Apr 24, 2015 #1
    1. The problem statement, all variables and given/known data
    Solve in ##\mathbb{Z}^2## the equation ## x^2 -y^2-x+3y = 30 ##

    2. Relevant equations


    3. 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) }
     
  2. jcsd
  3. Apr 24, 2015 #2

    Svein

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    Science Advisor

    Seems OK to me.
     
  4. Apr 24, 2015 #3
    Thanks, there was a typo, I forgot ##\pm 14## in Div(28) but I took it into account
     
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