Jacobi symbol $\left(\frac{(n+1)/2}{n}\right)$

Question

Let $(\frac{a}{n})_J$ be Jacobi symbol defined by \begin{equation} \left(\frac{a}{n}\right)_J=\left(\frac{a}{p_1}\right)^{e_1}\left(\frac{a}{p_2}\right)^{e_2}\cdots\left(\frac{a}{p_k}\right)^{e_k} \end{equation}

where $n=p_1^{e_1}p_2^{e_2}\cdots p_k^{e_k}$ is the prime factorization of $n$ and $(\frac{a}{p})$ is Legendre symbol.

I want to prove that for odd integer $n$ \begin{equation} \left(\frac{(n+1)/2}{n}\right)_J=\left\{ \begin{array}{ll} (-1)^{\frac{n-1}{4}}, & n=1 ~mod~~4 \\ (-1)^{\frac{n+1}{4}}, & n=3 ~mod~~4.\ \end{array} \right. \end{equation}

Thanks in advance. Happy new year.

Answer 1

Hint: Use known information about the value of $\left(\frac{a}{n}\right)_J\left(\frac{b}{n}\right)_J$, in the case $a=2$, $b=(n+1)/2$. Use also known information about the value of $\left(\frac{2}{n}\right)_J$.