The Finiteness Problem

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, for a given Diophantine equation, whether the set of its integer solutions is finite or infinite. In the latter case, it also investigates whether an equation has a solution with all variables arbitrarily large. The chapter starts with trivial examples, but then discusses equations that require more interesting methods, such as the law of quadratic reciprocity, binary quadratic forms, and Pell’s equation.