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Omni-MATH / For each positive integer n, denote by ω(n) the number of distinct prime divisors of n (for example, ω(1)=0 and ω(12)=2). Find all polynomials P(x) with integer coefficients, such …

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problem

For each positive integer nn, denote by ω(n)\omega(n) the number of distinct prime divisors of nn (for example, ω(1)=0\omega(1)=0 and ω(12)=2\omega(12)=2). Find all polynomials P(x)P(x) with integer coefficients, such that whenever nn is a positive integer satisfying ω(n)>20232023\omega(n)>2023^{2023}, then P(n)P(n) is also a positive integer with ω(n)≥ω(P(n)).\omega(n)\ge\omega(P(n)). Greece (Minos Margaritis - Iasonas Prodromidis)
Plain-text mathematical notation (without MathML)
For each positive integer n, denote by ω(n) the number of distinct prime divisors of n (for example, ω(1)=0 and ω(12)=2). Find all polynomials P(x) with integer coefficients, such that whenever n is a positive integer satisfying ω(n)>2023²⁰²³, then P(n) is also a positive integer with
ω(n)≥ω(P(n)).

Greece (Minos Margaritis - Iasonas Prodromidis)
Original LaTeX notation
For each positive integer $n$, denote by $\omega(n)$ the number of distinct prime divisors of $n$ (for example, $\omega(1)=0$ and $\omega(12)=2$). Find all polynomials $P(x)$ with integer coefficients, such that whenever $n$ is a positive integer satisfying $\omega(n)>2023^{2023}$, then $P(n)$ is also a positive integer with
\[\omega(n)\ge\omega(P(n)).\]

Greece (Minos Margaritis - Iasonas Prodromidis)

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Source and history

task_id
4fde9317-38d8-5929-9a97-bdd01546dae5
task_key
test--4fde9317-38d8-5929-9a97-bdd01546dae5
task_revision_id
1
config / split
/ test

Official source

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