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

Ca,n = {(x,y) : x2 + y2 a(modn),0 x,y n 1}

and the modular hyperbola

n = {(x,y) : xy 1(modn),0 x,y n 1}.

We provide explicit formulas for the cardinality of the sets

{amodn : Ca,n n}and{amodn : Ca,n}.

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