## Saturday, March 09, 2019

### [jozupeos] Small factors of central binomial coefficients

We investigate how many times some small primes {2,3,5,7} divide central binomial coefficients c(n)=binomial(2*n,n).

For example, c(6) = 12 choose 6 = 924 = 2^2 * 3^1 * 5^0 * 7^1 * (higher factors), so the exponents are 2,1,0,1.  We write zero as period in the table below to distinguish zeroes from nonzeros.

2^x 3^x 5^x 7^x . . . . 1 . . . 1 1 . . 2 . 1 . 1 . 1 1 2 2 . 1 2 1 . 1 3 1 . . 1 2 1 . 2 . 1 . 2 . . . 3 1 . 1 2 . . 1 3 . 2 1 3 3 2 . 4 2 1 . 1 2 1 . 2 3 1 . 2 1 2 1 3 1 2 1 2 2 1 1 3 1 1 . 3 1 1 . 4 3 2 . 2 2 2 . 3 2 . 2 3 3 . 2 4 . . 2 3 . 1 1 4 1 1 1 4 . . 1 5 . . 1 1 2 . 2

It seems p does not divide c(p^n) for p>2.  For p=2, 2 divides c(2^n) exactly once, probably from the 2 in the definition of c(n).

Here are some bar graphs revealing a fractal structure.  The horizontal axis goes from n=0 to 1199, that is, from (0 choose 0) to (2398 choose 1199); the vertical axis is stretched by a factor of 50.