Equations of the form are called *Mordell equations*. In this post we’re going to prove that the equation has no integer solutions, using (with one exception) nothing more complicated than congruences.

**Theorem**: There are no integer solutions to the equation .

**Proof**.

**Case 1**. First, suppose that there is a solution in which *x* is even. Since , . Looking at the equation modulo 8, then, we have

.

However, if we square the residues and reduce them modulo 8 we have . So there is no integer *y* that when squared is congruent to 7 modulo 8. Therefore, there is no solution to when *x *is even.

**Case 2**. Now, suppose there is a solution in which *x* is odd. If so, then we have or .

If then we have . However, if we square the residues and reduce them modulo 4 we get . So there is no integer *y* that when squared is congruent to 2 modulo 4. Thus there is no solution to when .

Now, suppose . Applying some algebra to the original equation, we have

.

By assumption, . If has prime factors that are all equivalent to 1 modulo 4, then their product (i.e., ) would be equivalent to 1 modulo 4. Thus has at least one prime factor *q* that is congruent to 3 modulo 4.

However, if , then . This means that . However, thanks to Euler’s criterion, only odd primes *q* such that can have a solution to . Since , has no solution. Thus there is no solution to when , either.

Since we’ve covered all cases, there can be no solution in integers to the Mordell equation .