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:
y2 = x3 + 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