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Double factorial divisibility

Source: Philippine Mathematical Olympiad 2024 P2

February 24, 2024

factorialnumber theory

Problem Statement

Let

0!!=1!!=1

and

n!!=n\cdot (n-2)!!

for all integers

n\geq 2

. Find all positive integers

n

such that

\dfrac{(2^n+1)!!-1}{2^{n+1}}

is an integer.

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