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Omni-MATH / Let n be a positive integer. Given an integer coefficient polynomial f(x), define its [i]signature modulo n[/i] to be the (ordered) sequence f(1),…,f(n) modulo n. Of the n^(n) such…

Problem

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problem

Let

n

be a positive integer. Given an integer coefficient polynomial

f(x)

, define its [i]signature modulo

n

[/i] to be the (ordered) sequence

f(1), \ldots , f(n)

modulo

n

. Of the

n^n

such

n

-term sequences of integers modulo

n

, how many are the signature of some polynomial

f(x)

if a)

n

is a positive integer not divisible by the square of a prime. b)

n

is a positive integer not divisible by the cube of a prime.

Plain-text mathematical notation (without MathML)

Let n be a positive integer. Given an integer coefficient polynomial f(x), define its [i]signature modulo n[/i] to be the (ordered) sequence f(1),…,f(n) modulo n. Of the n^(n) such n-term sequences of integers modulo n, how many are the signature of some polynomial f(x) if
a) n is a positive integer not divisible by the square of a prime.
b) n is a positive integer not divisible by the cube of a prime.

Original LaTeX notation

Let $ n$ be a positive integer. Given an integer coefficient polynomial $ f(x)$, define its [i]signature modulo $ n$[/i] to be the (ordered) sequence $ f(1), \ldots , f(n)$ modulo $ n$. Of the $ n^n$ such $ n$-term sequences of integers modulo $ n$, how many are the signature of some polynomial $ f(x)$ if
a) $ n$ is a positive integer not divisible by the square of a prime.
b) $ n$ is a positive integer not divisible by the cube of a prime.

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task_id: a70848ca-5d46-52da-81d5-19d9d585d712
task_key: test--a70848ca-5d46-52da-81d5-19d9d585d712
task_revision_id: 4
config / split: / test

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