I have to prove this statement for my class, but I have run into an issue. When I choose $n=15$, for example, and if I choose $p=3$, I get \begin{align} \sum_{\substack{1 \leq m \leq 8 \\[1pt] \gcd(m,15)=1}} \biggl(\frac{m}{3\strut}\biggr) &= \biggl(\frac{1}{3\strut}\biggr) + \biggl(\frac{2}{3\strut}\biggr) + \biggl(\frac{4}{3\strut}\biggr) + \biggl(\frac{7}{3\strut}\biggr) + \biggl(\frac{8}{3\strut}\biggr) \\ &= 1 - 1 + 1 + 1 - 1 = 1 \end{align} not $0$. What might I be doing wrong?
2026-03-27 21:20:35.1774646435
Prove that if $n$ is odd and $p \mid n$, then $\sum_{m=1,\, \gcd(m,n)=1}^{\varphi(n)} (\frac{m}{p})=0$
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