#### Vol. 3, No. 2, 2010

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

### Sara Hanrahan and Mizan Khan

Vol. 3 (2010), No. 2, 171–182
##### Abstract

Consider the modular circle

${\mathsc{C}}_{a,n}=\left\{\left(x,y\right):{x}^{2}+{y}^{2}\equiv a\phantom{\rule{0.3em}{0ex}}\left(mod\phantom{\rule{0.3em}{0ex}}n\right),\phantom{\rule{0.3em}{0ex}}0\le x,y\le n-1\right\}$

and the modular hyperbola

${\mathsc{ℋ}}_{n}=\left\{\left(x,y\right):xy\equiv 1\phantom{\rule{0.3em}{0ex}}\left(mod\phantom{\rule{0.3em}{0ex}}n\right),\phantom{\rule{0.3em}{0ex}}0\le x,y\le n-1\right\}.$

We provide explicit formulas for the cardinality of the sets

$\left\{a\phantom{\rule{0.3em}{0ex}}mod\phantom{\rule{0.3em}{0ex}}n:{\mathsc{C}}_{a,n}\cap {\mathsc{ℋ}}_{n}\ne \varnothing \right\}\phantom{\rule{1em}{0ex}}and\phantom{\rule{1em}{0ex}}\left\{a\phantom{\rule{0.3em}{0ex}}mod\phantom{\rule{0.3em}{0ex}}n:{\mathsc{C}}_{a,n}\ne \varnothing \right\}.$

##### Keywords
cardinality, value sets, modular circle, modular hyperbola
##### Mathematical Subject Classification 2000
Primary: 11A07, 11A25