Does the diophantine equation $$ ax+by+cxy+d=0 $$ always have solutions ?
2026-03-31 08:15:55.1774944955
What are the general solutions of the Diophantine equation $ ax+by+cxy+d=0 $
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I assume $c \ne 0$. Write the equation as $$ (cx + b) (cy + a) = ab - cd $$ So there are integer solutions iff you can factor $ab - cd$ into factors congruent to $a$ and $b$ mod $c$.