remainder of the product of the relatively prime numbers

whodsow

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

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

Thanks and Regards.

CRGreathouse

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

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 [tex]\phi(m)[/tex] then [tex]B{\equiv}-1[/tex], and if [tex]B{\equiv}1[/tex], there must be that 4 divides [tex]\phi(m)[/tex]

whodsow

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

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CRGreathouse Recognitions: Homework Help Science Advisor	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: 2 1 3 -1 4 -1 5 -1 6 -1 7 -1 8 1 9 -1 10 -1 11 -1 12 1 13 -1 14 -1 15 1 16 1 17 -1 18 -1 19 -1 20 1 Does that help?

whodsow	remainder of the product of the relatively prime numbers Media Man told me it is a generalization of Wilson's Theorem(http://mathworld.wolfram.com/WilsonsTheorem.html) Thanks.