Proving the Equation: 1/p c(p,n) = (-1)^{n-1}/n (mod p)

[ehrenfest]

Homework Statement

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 :(

Homework Equations

The Attempt at a Solution

 

[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.

 

[ehrenfest]

What do you mean "negative representatives"?

 

[Gokul43201]

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

 

[matt grime]

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

 

[NateTG]

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

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

 

[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.

 

[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?