Graham’s central-binomial-coefficient conjecture

Prove that infinitely many central binomial coefficients are not divisible by 105=3\cdot5\cdot7, equivalently, establish that infinitely many integers have only digits 0 and 1 in base 3, only digits 0, 1, and 2 in base 5, and only digits 0, 1, 2, and 3 in base 7.

Background

The conjecture is formulated through the base expansions of integers and the divisibility of central binomial coefficients. Using the identity relating the p-adic valuation of \binom{2n}{n} to the sum of digits s_p(n), nondivisibility by 105 is equivalent to simultaneous restrictions on the allowed digits in bases 3, 5, and 7. The paper describes this as a long-standing conjecture.