binomial coefficient modulo a prime

by erszega
Tags: binomial, coefficient, modulo, prime
erszega is offline
Jun11-06, 02:37 AM
P: 36
A question:

Let bin(a,b) denote the binomial coefficient a! / ( b! (a - b)! ).

Is it true that

bin( 2p, p ) = 2 (mod p) if p is prime and p>=3 ?
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matt grime
matt grime is offline
Jun11-06, 04:22 AM
Sci Advisor
HW Helper
P: 9,398
Yes, it's fermat's little theorem: x^p=x mod p, for p a prime, hence

(1+x)^2p = (1+x^p)^2 = 1+2x^p+x^{2p} mod p

note your requirement on p>=3 is not necessary. 4 choose 2 =6 whcih is congruent to 2 mod 2 as well.

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