True or False(2)

Number Theory Level 2

Is it true or false that for any odd prime $p$ the following exists: $1^2\times3^2\times5^2\times\dots\times(p-2)^2\equiv(-1)^{\frac{p+1}{2}} \mod p$

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1 solution

Arousse Fares
Aug 11, 2018

We first put some order in this messy equation and exploit the fact that $p+n \equiv n \mod p$ :

$A(p) = 1^{2} \times 3^{2} \times ... \times (p-2)^{2} = 1 \times (p-2) \times 3 \times (p-4) \times ... \times (p-(p-1)) \equiv 1 \times (-2) \times 3 \times (-4) \times ... \times (p-2) \times (-(p-1)) \mod p$

Which means that if there is an odd number of even numbers then $A(p) = - (p-1)!$ and if there is an even number of even numbers then $A(p) = (p-1)!$

According to Wilson's theorem : $(n-1)! \equiv -1 \mod n$ if and only if $n$ is prime.

And what a coincidence because we do have a prime number !!

The rest is easy. There are two cases : either you have an even number of even numbers or an odd number of even numbers. You just have to study the two scenarios with $\mod 4$ ...

Thank you for the problem ! It taught me Wilson's theorem which is quite a powerful tool.

Thank you, nice solution.

Hana Wehbi - 2 years, 10 months ago

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