Modulus Predictions for Binomial Expansion Coefficients?

[coolul007]

I am trying to predict the modulus without really doing the expansion. Therefore I'm in a snag with actually computing vs. only computing what I think I need. Here's the assumption I am Making:

n C r == 0 mod (n-1) for all r > 1

n C r are the coefficients of the binomial expansion. My assumption is that all the middle terms are zero and that only the:

x^n + nx^(n-1) + 0 +... +0 + nx +1 need to be computed mod(n-1)

My theoretical results are conflicting with the actual.

 

[robert2734]

The middle terms are 0 mod(n-1) if and only if n-1 is prime.

 

[coolul007]

The middle terms are 0 mod(n-1) if and only if n-1 is prime.

Is this still a true statement for n-1 not prime? I am having trouble as the expansion of n! == 0 mod(n-1), can you explain? Thanx

n C r == 0 mod (n-1) for all r > 1

 

[robert2734]

say n-1=6 then the expansion is x^7+7x^6+21x^5+35x^4+35x^3+21x^2+7x+1. The middle terms are not multiples of 6.

if n-1=5 x^6+5x^5+15x^4+20x^3+15x^2+5x+1, the middle terms are multiples of 5 because 5 is prime.

 

[coolul007]

I kind of figured out the why. When n-1 is prime there exists only one number to divide into. When n-1 is composite there exist the prime factors of n-1 plus there multiples to divide the n-1 composite in the "numerator" thus leaving no value for n-1 to divide into. It then becomes coprime to the rest of the coefficient. Well, back to the drawing board.