Solutions to Elementary Methods for Solving Diophantine Equations

Abstract

1. Solve the following equation in integers x, y :

$$x^{2} + 6xy + 8y^{2} + 3x + 6y = 2.$$

Solution. Write the equation in the form

$$(x + 2y)(x + 4y) + 3(x + 2y) = 2\quad{\rm or}\quad(x +2y)(x +4y + 3) = 2.$$

We obtain the solutions (0,−1), (3,−2), (3,−1), (6,−2).