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Number Theory Congruence equation

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Poirot

Banned
Feb 15, 2012
250
Let p be prime and a be between 1 and p-1. Show the binomial coefficent (p-1)C(a) satifies

(p-1)C(a) =(-1)^a mod(p).

(p-1)C(a) =$\frac{(p-1)!}{a!(p-1-a)!}$ so we can apply wilson's theorem which says
(p-1)!=-1 (modp)
 

caffeinemachine

Well-known member
MHB Math Scholar
Mar 10, 2012
834
Re: congruence equation

Let p be prime and a be between 1 and p-1. Show the binomial coefficent (p-1)C(a) satifies

(p-1)C(a) =(-1)^a mod(p).

(p-1)C(a) =$\frac{(p-1)!}{a!(p-1-a)!}$ so we can apply wilson's theorem which says
(p-1)!=-1 (modp)
Let $x\binom{p-1}{a+1}\equiv \binom{p-1}{a}\pmod{p}$.

Then we have $\frac{x(p-1)!}{(p-a-2)!(a+1)!}\equiv \frac{(p-1)!}{(p-a-1)!a!}\pmod{p}$.

Cancel things out (why can that be done?), you get $x\equiv -1\pmod{p}$

Now apply induction.