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Omni-MATH / The geometric mean of any set of m non-negative numbers is the m -th root of their product. (i) For which positive integers n is there a finite set S_(n) of n distinct positive i…

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problem

The geometric mean of any set of mm non-negative numbers is the mm -th root of their product. (i)\quad (\text{i})\quad For which positive integers nn is there a finite set SnS_n of nn distinct positive integers such that the geometric mean of any subset of SnS_n is an integer? (ii)\quad (\text{ii})\quad Is there an infinite set SS of distinct positive integers such that the geometric mean of any finite subset of SS is an integer?
Plain-text mathematical notation (without MathML)
The geometric mean of any set of m non-negative numbers is the m -th root of their product.
 (i)  For which positive integers n is there a finite set S_(n) of n distinct positive integers such that the geometric mean of any subset of S_(n) is an integer?
 (ii)  Is there an infinite set S of distinct positive integers such that the geometric mean of any finite subset of S is an integer?
Original LaTeX notation
The geometric mean of any set of $m$ non-negative numbers is the $m$ -th root of their product.
$\quad (\text{i})\quad$ For which positive integers $n$ is there a finite set $S_n$ of $n$ distinct positive integers such that the geometric mean of any subset of $S_n$ is an integer?
$\quad (\text{ii})\quad$ Is there an infinite set $S$ of distinct positive integers such that the geometric mean of any finite subset of $S$ is an integer?

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Source and history

task_id
2a3953cc-4cf8-56a0-b26f-3461429e6380
task_key
test--2a3953cc-4cf8-56a0-b26f-3461429e6380
task_revision_id
1
config / split
/ test

Official source

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