# Nonlinear Diophantine Equations

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 .

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 .

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 .

Thus, .

Subtracting the two equations gives

The second equation gives . Plugging this into gives

This equation has no real solutions.

Case 4. and .

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.

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