Does $x^{11} - 23y =3$ have integer solutions?

Question

I am trying to figure out whether or not $x^{11} - 23y =3$ has any integer solutions. I am not trying to find the solutions, but rather find out if any exists.

Here is how I have approached it so far. This reminded me of Pell's equation, $x^2 - Dy^2 = 1$ which always has solutions. However, I don't think this can really help me. My next thought was to use some argument that involves divisibility to possibly show that there are no solutions, but I'm not really sure how.

Answer 1

There are no solutions. It follows from Fermat's Little Theorem that the only 11th powers mod 23 are 0,1 and -1. To see this, Assume $x \not \equiv 0$ (mod 23). Then $x^{22} \equiv 1$ (mod 23) by Fermat. And thus $(x^{11}-1)(x^{11}+1) \equiv 0$ (mod 23). So $23|x^{11}+1$ or $23|x^{11}-1$ (since 23 is prime). So, $x^{11} \pm 1$ (mod 23). Since $\pm 1 \not \equiv 3$ (mod 23). So there are no solutions of the original equation.

Answer 2

Hint: Consider the set of 11-th powers mod 23.

Answer 3

Fermat's little theorem tells you that if $x$ is not a multiple of $23$, then $x^{22} \equiv 1 \mod{23}$. From this you can deduce that $x^{11} \equiv \pm 1 \mod{23}$.