Probably it would help if you spelled the word "Diophantine" - the
equations are named after Diophantus of Alexandria, a famous Greek
mathematician of antiquity.
A Diophantine equation is one equation in at least two variables, say
x and y, whose solutions (x,y) are required to be whole numbers
(integers). Some such equations have no solutions. Some have a finite
number. Some have infinitely many. The example of the Pythagorean
equation is one:
a^2 + b^2 = c^2.
If you are asked for integer solutions, this is a Diophantine
equation. You already know some solutions, I am sure:
(a,b,c) = (0,0,0), (0,1,1) (0,1,-1), (0,-1,1), (0,-1,-1), (1,0,1),
(1,0,-1), (-1,0,1), (-1,0,-1),
for example. You can see that for this equation, you can assume that
a, b, and c are all positive. From that solution you can obtain others
by changing signs of some or all of a, b, and c. Furthermore, you can
assume a <= b, since you can take any solution and swap a and b to get
another. In this case there are infinitely many solutions.
Another Diophantine equation might be 5*x + 7*y = 57. This, too, has
an infinite number of solutions, and they are given by x = 3 + 7*t,
y = 6 - 5*t, where t is any integer.
Hehe, neva' mind
