Posted by lin on .
Find the smallest n such that for some prime p, at least 20 of the numbers 1,2,...,n are quadratic nonresidues
modulo p.

heeeeeeeelp math 
exactly,
I suppose this question isn't live anymore. Anyways next time please don't post brilliant problems (:
Well, here's a few hints for you to work out and be on the right track:
Step 1: quadratic reciprocity and CRT
Step 2: incorporate dirichlet's theorem into this.
Well, you can then see obviously that {3,5,6,7,10,11,12,13,14,17,19,20,22,23,24,26,27,28,29,31}. Now just try to prove that the desired answer is minimal. This is simple. Show that 2 and 3 MUST be quadratic residues. 
heeeeeeeelp math 
exactly,
Ok to clarify, the show that 2 and 3 must be quadratic residues part is to find the minimal.