In general, solving a nonlinear Diophantine equation can be very difficult. In this section, we'll look at some examples of solving such an equation, and showing that such an equation can't be solved.

* Example.* Find all pairs of nonnegative
integers such that

Case 1. and .

Adding the two equations gives

Thus, .

Subtracting the two equations gives

The second equation gives . Plugging this into gives

gives and gives . The two solutions in this case are and .

Case 2. and .

Adding the two equations gives

Thus, .

Subtracting the two equations gives

The second equation gives . Plugging this into gives

gives and gives . The two solutions in this case are and .

Case 3. and .

Adding the two equations gives

Thus, .

Subtracting the two equations gives

The second equation gives . Plugging this into gives

This equation has no real solutions.

Case 4. and .

Adding the two equations gives

Thus, .

Subtracting the two equations gives

The second equation gives . Plugging this into gives

This equation has no real solutions.

The solutions are , , , and .

* Example.* Prove that the following Diophantine
equation has no solutions:

I reduce the equation mod 5 to obtain

I construct a table of squares mod 5:

This shows that 2 is not a square mod 5. Hence, the original Diophantine equation has no solutions.

Copyright 2019 by Bruce Ikenaga