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> Let $p$ be an odd prime. Then among the integers $\{1,2,3,\cdots,p-1\}$ exactly half are quadratic residues modulo $p$.
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I believe the above proposition. 

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> Let $n$ be an odd, square free integer. How many integers $\{1,2,3,\cdots,n-1\}$ are quadratic residues mod $n$.
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There are exactly $\phi(n)$ integers that are relatively prime to $n$. Further, we can factor $n=P\_1\cdots P\_N$. I keep thinking that the number of integers will be less than $\phi(n)/2$ or $(n-1)/2$. I'm not sure if I am right or how to prove it.