# Number theory problem

## 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 :(

## Answers and Replies

NateTG
Science Advisor
Homework Helper
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.

What do you mean "negative representatives"?

Gokul43201
Staff Emeritus
Science Advisor
Gold Member
-(p-1)/2, -(p-2)/2,..., -1 for odd p

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matt grime
Science Advisor
Homework Helper
What is 1/n, or -1/n mod p supposed to mean?

NateTG
Science Advisor
Homework Helper
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
Science Advisor
Homework Helper
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.

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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?