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OlympiadBench / 1929 / Find all pairs (m,n) of positive integers satisfying the equation

Problem

Answer published by the source. Consult the official source to check your work against its answer.

question

Find all pairs (m,n)(m, n) of positive integers satisfying the equation $$ \left(2^{n}-1\right)\left(2^{n}-2\right)\left(2^{n}-4\right) \cdots\left(2^{n}-2^{n-1}\right)=m ! \tag{1} $$
Plain-text mathematical notation (without MathML)
Find all pairs (m,n) of positive integers satisfying the equation

$$
\left(2^{n}-1\right)\left(2^{n}-2\right)\left(2^{n}-4\right) \cdots\left(2^{n}-2^{n-1}\right)=m !
\tag{1}
$$
Original LaTeX notation
Find all pairs $(m, n)$ of positive integers satisfying the equation

$$
\left(2^{n}-1\right)\left(2^{n}-2\right)\left(2^{n}-4\right) \cdots\left(2^{n}-2^{n-1}\right)=m !
\tag{1}
$$

answer type

Tuple

is multiple answer

true

language

English

question type

Open-ended

subject

Math

Discussion

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Source and history

upstream_id / Original problem ID
1929
task_id
5cdf216f-64fb-5d0d-9d4b-f0e441ac30e3
task_key
OE~5fTO~5fmaths~5fen~5fCOMP--train--1929
task_revision_id
4
config / split
OE_TO_maths_en_COMP / train

Official source

initial import