Equations in Two Variables

The size of a polynomial with integer coefficients is computed by replacing each coefficient by its absolute value and evaluating the resulting polynomial at 2. This chapter investigates two-variable Diophantine equations according to their size. It starts with elementary methods like Vieta jumping and Pell’s equations, but also discusses much more advanced topics like computing integral points on elliptic and superelliptic curves. For advanced topics, only references are presented without proofs, to make the chapter accessible to undergraduate students. By combining elementary and advanced methods, the chapter establishes that all two-variable equations up to the size 27 can be solved. The smallest unsolved equations are left as open questions.