Question from the proof in euler's forumla

lifom

Let b1,b2,...bn be the integers between 1 and m that are relative prime to m (including 1), and let B = b1*b2*...*bn be their product. The quantity B came up during the proof of euler formula. a^n = 1 (mod m), where n is number of integers between 1 and m that relative prime to m.

How can I show B=1 (mod m ) or B = -1(mod m)?

micromass

Have you seen Wilson's theorem? This states that (p-1)!=-1 (mod p) for prime p.

Can you adjust the proof so that it also works in this case?

lifom

I have tried, but fail in the last step:
In Wilson's proof, the number satisfy a^2=1(mod p) -> (a-1)(a+1) = p*n. Since 0<=a<p
a= 1 or p-1 only. So for 2,3,...,p-2, I can group them into pairs without repeat such that the pair product = 1 (mod p)...

But in my problem, the number satisfy a^2=1(mod m) does not imply a=1 or m-1 since m is a composite number. (e.g. 3^2=1(mod 8))

I also modify my proof. Consider B^2=b1^2*b2^2*...bn^2. I have showed that for each bi, there exist unique bj such that bi*bj=1 (mod m) (similar to Wilson’s proof). Although we don't know bi is equal to bj or not.

So I divide the number into 2 groups,
1st group: bi*bj=1 (mod m) for i <>j
2nd group: bi*bj=1 (mod m) for i = j

So B^2 = 1 (mod m) but the same problem occurs,
It cannot be concluded that B= 1 or -1(mod m)

micromass

Well, you could show the following:

For each b_i, there exists a b_j such that b_ib_j=1 (mod m). If b_i happens to equal b_j, then there exists a b_r such that b_ib_r=-1 (mod m).

So you can group the product into pairs again. Some pairs will multiply to 1 and some will multiply to -1.

lifom

I get it! Thanks.

ramsey2879

If b_j^2 = 1 mod m then b_j *(m -b_j) = -1 mod m