A Diophantine Equation is characterized not by the shape of the equation but by the fact that one is interested only in solutions in integers (whole numbers) or rational numbers. Typically, it will have more than one variable. For example, xy = x + y + 2 can be looked at as a diophantine equation and it has only the solutions (x,y)=(2,4),(4,2),(0,-2),(-2,0) in integers. The equation 2x+4y = 1 is also a diophantine equation, and in this case it has no integral solutions, since the left hand side is even if x and y are integers and 1 is not even.

Historically, the first record of the study of Diophantine Equations is a Babylonian tablet, dating from before 1600 B.C., which lists integer solutions of the Pythagorean equation x²+y²=z², which leads to right-angled triangles with integral sides. Diophantine Equations were also studied by the Greeks, in particular...