Importance: Medium ✭✭
Author(s): Graham, Ronald L.
Keywords:
Recomm. for undergrads: yes
Prize: $1,000 each
Posted by: maxal
on: August 5th, 2007
Problem  (1)   Prove that there exist infinitely many positive integers $ n $ such that $$\gcd({2n\choose n}, 3\cdot 5\cdot 7) = 1.$$
Problem  (2)   Prove that there exists only a finite number of positive integers $ n $ such that $$\gcd({2n\choose n}, 3\cdot 5\cdot 7\cdot 11) = 1.$$

The binomial coefficient $ {2n\choose n} $ is not divisible by prime $ p $ iff all the base-$ p $ digits of $ n $ are smaller than $ \frac{p}{2}. $

It has been conjectured that 1, 2, 10, 3159, and 3160 are the only positive numbers for which $ \gcd({2n\choose n}, 3\cdot 5\cdot 7\cdot 11) = 1 $ holds.


* indicates original appearance(s) of problem.

Reply

Comments are limited to a maximum of 1000 characters.
More information about formatting options