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A. 787

Part of KoMaL A Problems 2020/2021

Problems(1)

Messing with Binomial Coefficients

Source: KöMaL A. 787 (November 2020), Universal Mathematical Olympiad

7/27/2020
Let pnp_n denote the nthn^{\text{th}} prime number and define an=⌊pnν⌋a_n=\lfloor p_n\nu\rfloor for all positive integers nn where ν\nu is a positive irrational number. Is it possible that there exist only finitely many kk such that (2akak)\binom{2a_k}{a_k} is divisible by pi10p_i^{10} for all i=1,2,…,2020?i=1,2,\ldots,2020?
Proposed by Superguy and ayan.nmath
number theorybinomial coefficientsDivisibilitynumber theory proposedKomal prob