Use for questions about finding integer or rational solutions to polynomial equations.
Use this tag for questions about finding integer, or perhaps rational, solutions to polynomial equations.
Diophantine equations are named after Diophantus of Alexandria, a third century Greek mathematician.
An example of a Diophantine equation is to find all quadruples of integers $(w,x,y,z)$ such that $$w^2+x^2=3(y^2+z^2).$$
Solving Diophantine equations often involves other areas of mathematics such as congruences, linear algebra, inequalities, forms (e.g., binary quadratic forms), and elliptic curves. Special solution methods include comparing divisors, considering orders of magnitude, Fermat's method of descent, and finding intersections of curves with lines of rational slope through a known rational point.