Number theory problem

1. Oct 30, 2007

ehrenfest

1. The problem statement, all variables and given/known data
Prove that

$$\frac{1}{p} c(p,n) = (-1)^{n-1}/n (mod p)$$

I expanded that combination in every way I could think and I tried to use Wilson's Theorem and I couldn't get :(

2. Relevant equations

3. The attempt at a solution

2. Oct 30, 2007

NateTG

That's p choose n, right?

Try writing the LHS out as a fraction with the stuff in the numerator as negative representatives. It should nicely cancel to give the result.

3. Oct 30, 2007

ehrenfest

What do you mean "negative representatives"?

4. Oct 30, 2007

Gokul43201

Staff Emeritus
-(p-1)/2, -(p-2)/2,..., -1 for odd p

Last edited: Oct 30, 2007
5. Oct 30, 2007

matt grime

What is 1/n, or -1/n mod p supposed to mean?

6. Oct 31, 2007

NateTG

Usually those would be the multiplicative inverses of n and -n respectively.

7. Oct 31, 2007

matt grime

Usually? I beg to differ. Writing 1/n would indicate that the OP hasn't grasped what's going on. As would the fact there is an equals sign. I can't think of anyone who writes 1/2 mod 3 and not -1 0 it is incredibly bad notation. There is a difference from what I infer and what the OP ought to have written.

Last edited: Oct 31, 2007
8. Nov 1, 2007

ehrenfest

What does OP stand for? Is that me?

I just realized that my book my book defines congruence as

$$x \equiv y \mod p$$

when x-y is a rational number whose numerator, in reduced form, is divisible by p.

So, it is like a generalized congruence or something...

Are there different rules for these generalized congruences?

I am not sure why what Gokul43201 wrote cancels nicely?