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Sequence of binomial coefficients modulo 2^k

Source: Kvant Magazine No. 8 2019 M2572

March 14, 2023
number theorybinomial coefficientsKvant

Problem Statement

Let kk be a fixed positive integer. Prove that the sequence (21),(42),(84),…,(2n+12n),…\binom{2}{1},\binom{4}{2},\binom{8}{4},\ldots, \binom{2^{n+1}}{2^n},\ldots is eventually constant modulo 2k2^k.
Proposed by V. Rastorguyev
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