# Equation with several variables

1. Apr 24, 2015

### geoffrey159

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. Apr 24, 2015

### Svein

Seems OK to me.

3. Apr 24, 2015

### geoffrey159

Thanks, there was a typo, I forgot $\pm 14$ in Div(28) but I took it into account

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