Competition problem — Number theory

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Number theory Difficulty 4.6 AIME Find the answer

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

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

None

Translate the text above into English, please retain the original text's line breaks and format, and output the translation result directly.

Note: The provided instruction is a meta-instruction and not part of the text to be translated. Since the text to be translated is "None", the translation is also "None". Here is the formatted output as requested:

None

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Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.