During my research I came across this question :

**Question:** What's the value of $x_p=(\dfrac{p-1}{2})! \mod p$ when $p>3$ is prime ?

*Remark:* It's easy to see $x_p^2 \mod p=(-1)^{\dfrac{p+1}{2}} \mod p$ with $x_p\in [1;p-1] \cap \mathbb N$

But it's not easy to find if $x_p \leq\dfrac{p-1}{2}$ or $x_p> \dfrac{p-1}{2}$

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