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Omni-MATH / A polynomial f∈Z[x] is called splitty if and only if for every prime p, there exist polynomials g_(p),h_(p)∈Z[x] with degg_(p),degh_(p)<degf and all coefficients of f−g_(p)h_(p) ar…

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problem

A polynomial f∈Z[x]f \in \mathbb{Z}[x] is called splitty if and only if for every prime pp, there exist polynomials gp,hp∈Z[x]g_{p}, h_{p} \in \mathbb{Z}[x] with deggp,deghp<degf\operatorname{deg} g_{p}, \operatorname{deg} h_{p}<\operatorname{deg} f and all coefficients of f−gphpf-g_{p} h_{p} are divisible by pp. Compute the sum of all positive integers n≤100n \leq 100 such that the polynomial x4+16x2+nx^{4}+16 x^{2}+n is splitty.
Plain-text mathematical notation (without MathML)
A polynomial f∈Z[x] is called splitty if and only if for every prime p, there exist polynomials g_(p),h_(p)∈Z[x] with degg_(p),degh_(p)<degf and all coefficients of f−g_(p)h_(p) are divisible by p. Compute the sum of all positive integers n≤100 such that the polynomial x⁴+16x²+n is splitty.
Original LaTeX notation
A polynomial $f \in \mathbb{Z}[x]$ is called splitty if and only if for every prime $p$, there exist polynomials $g_{p}, h_{p} \in \mathbb{Z}[x]$ with $\operatorname{deg} g_{p}, \operatorname{deg} h_{p}<\operatorname{deg} f$ and all coefficients of $f-g_{p} h_{p}$ are divisible by $p$. Compute the sum of all positive integers $n \leq 100$ such that the polynomial $x^{4}+16 x^{2}+n$ is splitty.

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Source and history

task_id
c3b1904b-6774-5e4a-8f7d-00206d19f3ec
task_key
test--c3b1904b-6774-5e4a-8f7d-00206d19f3ec
task_revision_id
1
config / split
/ test

Official source

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