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Omni-MATH / For a nonnegative integer n define rad(n)=1 if n=0 or n=1, and rad(n)=p₁p₂⋯p_(k) where p₁<p₂<⋯<p_(k) are all prime factors of n. Find all polynomials f(x) with nonnegative integer …

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problem

For a nonnegative integer nn define rad(n)=1\operatorname{rad}(n)=1 if n=0n=0 or n=1n=1, and rad(n)=p1p2⋯pk\operatorname{rad}(n)=p_1p_2\cdots p_k where p1<p2<⋯<pkp_1<p_2<\cdots <p_k are all prime factors of nn. Find all polynomials f(x)f(x) with nonnegative integer coefficients such that rad(f(n))\operatorname{rad}(f(n)) divides rad(f(nrad(n)))\operatorname{rad}(f(n^{\operatorname{rad}(n)})) for every nonnegative integer nn.
Plain-text mathematical notation (without MathML)
For a nonnegative integer n define rad(n)=1 if n=0 or n=1, and rad(n)=p₁p₂⋯p_(k) where p₁<p₂<⋯<p_(k) are all prime factors of n. Find all polynomials f(x) with nonnegative integer coefficients such that rad(f(n)) divides rad(f(n^(rad(n)))) for every nonnegative integer n.
Original LaTeX notation
For a nonnegative integer $n$ define $\operatorname{rad}(n)=1$ if $n=0$ or $n=1$, and $\operatorname{rad}(n)=p_1p_2\cdots p_k$ where $p_1<p_2<\cdots <p_k$ are all prime factors of $n$. Find all polynomials $f(x)$ with nonnegative integer coefficients such that $\operatorname{rad}(f(n))$ divides $\operatorname{rad}(f(n^{\operatorname{rad}(n)}))$ for every nonnegative integer $n$.

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Source and history

task_id
2b526f2d-7cc8-542b-aaa3-17e6d81cda9f
task_key
test--2b526f2d-7cc8-542b-aaa3-17e6d81cda9f
task_revision_id
1
config / split
/ test

Official source

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