Suppose we know $a,b,c,d,e,f,m\in\Bbb Z$ in $$(a^2c+b^2d)y+ab(vy)+(a^2e+b^2f)v=m$$ how do we find $v,y\in\Bbb Z$?
2026-03-28 09:44:14.1774691054
A bilinear diophantine problem
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Given $$ PQxy + Rx + Sy, $$ take $$ P^2 Q^2 xy + PQR x + PQSy. $$ Compare $$ (PQx +S) (PQy+R ) = P^2 Q^2 xy + PQR x + PQSy +RS$$
So, in your original problem, multiply both sides by $ab,$ then add something useful to both sides