The cardinality of the value sets modulo $n$ of $x^2+x^{-2}$ and $x^2+y^2$

Abstract

Consider the modular circle

$${\mathcal{C}}_{a,n}=\left\{\left(x,y\right):x^2+y^2\equiv a\left(modn\right),0\le x,y\le n-1\right\}$$

and the modular hyperbola

$${\mathcal{\mathscr{H}}}_n=\left\{\left(x,y\right):xy\equiv 1\left(modn\right),0\le x,y\le n-1\right\}.$$

We provide explicit formulas for the cardinality of the sets

$$\left\{amodn:{\mathcal{C}}_{a,n}\cap {\mathcal{\mathscr{H}}}_n\ne \varnothing \right\}and\left\{amodn:{\mathcal{C}}_{a,n}\ne \varnothing \right\}.$$