Home » Subject » Number Theory » Combinatorial N.T.
Importance: Medium ✭✭
Author(s): Graham, Ronald L.
Subject: Number Theory
» Combinatorial Number Theory
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.

Bibliography

P. Erdos, R.L. Graham, I.Z. Ruzsa and E.G. Straus "On the Prime Factor of $ 2n \choose n $." Math. Comp. 29 (1975), 83-92.

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."


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