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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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    $\begingroup$ See mathoverflow.net/questions/121678/… $\endgroup$
    – dhy
    Mar 29, 2019 at 13:19
  • $\begingroup$ He talks about the case $p \mod 4=3$ what about the other case ? $\endgroup$
    – Dattier
    Mar 29, 2019 at 13:29
  • $\begingroup$ See OEIS sequence A004055 for programs and the first 9999 terms. $\endgroup$
    – Robert Israel
    Mar 29, 2019 at 14:04
  • $\begingroup$ If you change the generator $b$ and replace it (for example) by $1/b$ then the sign you are interested in changes. So I think in the case p=1 mod 4, it does not really make sense to ask about this sign, as there is no canonical generator of $\mathbb{F}_p^{\times} $ $\endgroup$
    – François Brunault
    Mar 29, 2019 at 14:43
  • $\begingroup$ Nevertheless you could define a kind of canonical square root of -1 by using a decomposition $p=a^2+b^2$, say $a/b$ with $a <b $. But you see it depends on some choices, which could turn out to be arbitrary $\endgroup$
    – François Brunault
    Mar 29, 2019 at 14:51

2 Answers 2

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For the case $p\equiv 1\pmod{4}$, see the following paper of Chowla: On the class number of real quadratic fields. Proc. Nat. Acad. Sci. U.S.A. 47 (1961), 878. There the following is proved:

Let $h$ be the class number of $\mathbb{Q}(\sqrt{p})$, and let $\epsilon = (u+v\sqrt{p})/2$ be the fundamental unit of the corresponding ring of integers. Then $h$ is odd and $$ 2\cdot \frac{p-1}{2}! \equiv (-1)^{(h+1)/2} u \pmod{p}.$$

One can find an exposition of this result (and related material) in the last chapter of Pollack's Conversational Introduction to Algebraic Number Theory.

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See the paper by Mordell: The congruence (p−1/2)!≡±1 (mod p). Amer. Math. Monthly 68 1961 145–146.

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  • $\begingroup$ And $x_p=+-\sqrt{-1}$ when $p \mod 4=1$ $\endgroup$
    – Dattier
    Mar 29, 2019 at 13:20

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