ALG-B1-M10-P011 Difference of squares

Factor as \((x-y)(x+y)\) and keep track of the order of factors.

Let \(u=x-y\), \(v=x+y\). Then \(uv=35\), \(x=\frac{u+v}{2}\), \(y=\frac{v-u}{2}\). Both factors are odd, so all divisor pairs work. From \((u,v)=(1,35),(35,1),(5,7),(7,5)\) and the corresponding negative pairs, we get \((18,17),(18,-17),(6,1),(6,-1),(-18,-17),(-18,17),(-6,-1),(-6,1)\).

A Diophantine problem without announcing the method; the order of factors matters.