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Messing with Binomial Coefficients

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

July 27, 2020

number theorybinomial coefficientsDivisibilitynumber theory proposedKomal prob

Problem Statement

Let

p_n

denote the

n^{\text{th}}

prime number and define

a_n=\lfloor p_n\nu\rfloor

for all positive integers

n

where

\nu

is a positive irrational number. Is it possible that there exist only finitely many

k

such that

\binom{2a_k}{a_k}

is divisible by

p_i^{10}

for all

i=1,2,\ldots,2020?

Proposed by Superguy and ayan.nmath