binomial coefficient modulo a prime

erszega

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 ?

matt grime

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.