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Quadratic Residues

Source: Iran third round number theory exam 2015 - problem 3

September 9, 2015
number theoryprime numbersQuadratic Residues

Problem Statement

Let p>5p>5 be a prime number and A={b1,b2,…,bp−12}A=\{b_1,b_2,\dots,b_{\frac{p-1}{2}}\} be the set of all quadratic residues modulo pp, excluding zero. Prove that there doesn't exist any natural a,ca,c satisfying (ac,p)=1(ac,p)=1 such that set B={ab1+c,ab2+c,…,abp−12+c}B=\{ab_1+c,ab_2+c,\dots,ab_{\frac{p-1}{2}}+c\} and set AA are disjoint modulo pp.
This problem was proposed by Amir Hossein Pooya.
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