Some divisibility properties of binomial
and q-binomial coefficients
We first prove that if a has a prime
factor not dividing b then there are infinitely many
positive integers n such that
is not divisible by bn+1.
This confirms a recent conjecture of
Moreover, we provide some new divisibility properties of binomial
coefficients: for example, we prove that
by 6n-1, and that
is divisible by 66n-1, for all
positive integers n. As we show, the latter results are in fact
consequences of divisibility and positivity results for
quotients of q-binomial coefficients by q-integers,
generalising the positivity of q-Catalan numbers.
We also put forward several related conjectures.
The following versions are available:
Back to Christian Krattenthaler's