An

equation with any number of variables involving the basic arithmetic operations (

addition,

multiplication, and

exponentiation) considering only

integer solutions.

It has been proven that the problem of solving general Diophantine equations is undecidable. Some have no solutions, others have a finite number of solutions, some have an infinite number of solutions. Particular cases can be solved by one means or another, and Diophantine analysis is a major branch of mathematics and number theory. However, for many Diophantine equations, it is impossible to determine whether or not solutions even exist.

Example:

y^{2} = x^{3} + 1090

Solutions (probably incomplete):

x y
----------------------------
-9 19
-9 -19
-1 33
-1 -33
28,187,351 149,651,610,621
28,187,351 -149,651,610,621