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Omni-MATH / We say that a set S of integers is [i]rootiful[/i] if, for any positive integer n and any a₀,a₁,⋯,a_(n)∈S, all integer roots of the polynomial a₀+a₁x+⋯+a_(n)x^(n) are also in S. Fi…

Problem

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problem

We say that a set SS of integers is [i]rootiful[/i] if, for any positive integer nn and any a0,a1,⋯,an∈Sa_0, a_1, \cdots, a_n \in S, all integer roots of the polynomial a0+a1x+⋯+anxna_0+a_1x+\cdots+a_nx^n are also in SS. Find all rootiful sets of integers that contain all numbers of the form 2a−2b2^a - 2^b for positive integers aa and bb.
Plain-text mathematical notation (without MathML)
We say that a set S of integers is [i]rootiful[/i] if, for any positive integer n and any a₀,a₁,⋯,a_(n)∈S, all integer roots of the polynomial a₀+a₁x+⋯+a_(n)x^(n) are also in S. Find all rootiful sets of integers that contain all numbers of the form 2^(a)−2^(b) for positive integers a and b.
Original LaTeX notation
We say that a set $S$ of integers is [i]rootiful[/i] if, for any positive integer $n$ and any $a_0, a_1, \cdots, a_n \in S$, all integer roots of the polynomial $a_0+a_1x+\cdots+a_nx^n$ are also in $S$. Find all rootiful sets of integers that contain all numbers of the form $2^a - 2^b$ for positive integers $a$ and $b$.

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Source and history

task_id
d7683ff9-bef2-5330-bc17-e599909ccf61
task_key
test--d7683ff9-bef2-5330-bc17-e599909ccf61
task_revision_id
1
config / split
/ test

Official source

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