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at least one term of the sequence A, 2A,4A,8A,...,2^kA, ... is an integer

Source: Eotvos 1934 p1

September 10, 2024
number theoryIntegerfactorial

Problem Statement

Let nn be a given positive integer and A=1⋅3⋅5⋅...⋅(2n−1)2⋅4⋅6⋅...⋅2nA =\frac{1 \cdot 3 \cdot 5 \cdot ... \cdot (2n- 1)}{2 \cdot 4 \cdot 6 \cdot ... \cdot 2n} Prove that at least one term of the sequence A,2A,4A,8A,...,2kA,...A, 2A,4A,8A,...,2^kA, ... is an integer.
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