remainder of the product of the relatively prime numbers

**whodsow** · May28-09, 08:28 AM

Hi all, I had a problem, pls help me.

Let $LaTeX Code: b_1 < b_2 < \\cdots < b_{\\varphi(m)}$ be the integers between 1 and m that are relatively prime to m (including 1), of course, $LaTeX Code: \\varphi(m)$ is the number of integers between 1 and m that are relatively prime to m, and let $LaTeX Code: B = b_1b_2b_3{\\cdots}b_{\\varphi(m)}$ be their product.
Please find a pattern for when $LaTeX Code: B\\equiv1 ({\\bmod}\\ m)$ and when $LaTeX Code: B\\equiv-1 ({\\bmod}\\ m)$ .

Thanks and Regards.

**CRGreathouse** · May28-09, 09:57 AM

Code:

ff(m)=centerlift(prod(b=2,m,if(gcd(m,b)==1,b,1),Mod(1,m)))
for(m=2,20,print(m" "ff(m)))

in Pari/gp gives

Code:

Does that help?

**whodsow** · May29-09, 10:52 AM

Thank you, I've gotten the similar table.

Code:

but I still can't find the ppattern, and I have only found that if 4 does not divide $LaTeX Code: \\phi(m)$ then $LaTeX Code: B{\\equiv}-1$ , and if $LaTeX Code: B{\\equiv}1$ , there must be that 4 divides $LaTeX Code: \\phi(m)$

**whodsow** · May29-09, 11:52 AM

Media Man told me it is a generalization of Wilson's Theorem(http://mathworld.wolfram.com/WilsonsTheorem.html)
Thanks.