# Divisibility of central binomial coefficients

**Problem (1)**Prove that there exist infinitely many positive integers such that

**Problem (2)**Prove that there exists only a finite number of positive integers such that

The binomial coefficient is not divisible by prime iff all the base- digits of are smaller than

It has been conjectured that 1, 2, 10, 3159, and 3160 are the only positive numbers for which holds.

## Bibliography

Sequence A030979: Numbers n such that C(2n,n) is not divisible by 3, 5 or 7.

Andrew Granville. "The Arithmetic Properties of Binomial Coefficients."

* indicates original appearance(s) of problem.

### 2, 10, and 3159 are not valid for problem (2)

In the initial comment five values are stated to be not divisible by 3, 5, 7, and 11. That is true for 1 and 3160 but not for 2, 10, and 3159:

The last base-3-digit of 2 is 2. The last base-11-digit of 10 is 10. The last base-5-digit 3159 is 4.

Or check these facts:

2 and 3159 are not in the sequence A030979 (mentioned in the bibliography).

### Reference for problem (2)

It appears that the right reference for question 2 should be A151750 instead.

### Problem 1 solved?

Looks like Problem 1 was solved by http://arxiv.org/PS_cache/arxiv/pdf/1010/1010.3070v1.pdf

## It seems Problem (1) was

It seems Problem (1) was solved last year, see http://arxiv.org/abs/1010.3070

How about Problem 2?