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Omni-MATH / Find all polynomials P(x) of odd degree d and with integer coefficients satisfying the following property: for each positive integer n, there exists n positive integers …

Problem

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problem

Find all polynomials P(x)P(x) of odd degree dd and with integer coefficients satisfying the following property: for each positive integer nn, there exists nn positive integers x1,x2,…,xnx_1, x_2, \ldots, x_n such that 12<P(xi)P(xj)<2\frac12 < \frac{P(x_i)}{P(x_j)} < 2 and P(xi)P(xj)\frac{P(x_i)}{P(x_j)} is the dd-th power of a rational number for every pair of indices ii and jj with 1≤i,j≤n1 \leq i, j \leq n.
Plain-text mathematical notation (without MathML)
Find all polynomials P(x) of odd degree d and with integer coefficients satisfying the following property: for each positive integer n, there exists n positive integers x₁,x₂,…,x_(n) such that (1)/(2)<(P(x_(i)))/(P(x_(j)))<2 and (P(x_(i)))/(P(x_(j))) is the d-th power of a rational number for every pair of indices i and j with 1≤i,j≤n.
Original LaTeX notation
Find all polynomials $P(x)$ of odd degree $d$ and with integer coefficients satisfying the following property: for each positive integer $n$, there exists $n$ positive integers $x_1, x_2, \ldots, x_n$ such that $\frac12 < \frac{P(x_i)}{P(x_j)} < 2$ and $\frac{P(x_i)}{P(x_j)}$ is the $d$-th power of a rational number for every pair of indices $i$ and $j$ with $1 \leq i, j \leq n$.

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Source and history

task_id
8c060eaa-cb88-58fa-9019-31250115deef
task_key
test--8c060eaa-cb88-58fa-9019-31250115deef
task_revision_id
1
config / split
/ test

Official source

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