#### 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
##### Milestones
Received: 8 February 2009
Accepted: 21 June 2010
Published: 11 August 2010

Communicated by Filip Saidak
##### Authors
 Sara Hanrahan Department of Mathematics and Computer Science Eastern Connecticut State University Willimantic, CT 06226 United States Mizan Khan Department of Mathematics and Computer Science Eastern Connecticut State University Willimantic, CT 06226 United States