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MIT PRIMES/Art of Problem Solving

CROWDMATH 2024: Where Does the Goldbach Conjecture Hold?

Where Does the Goldbach Conjecture Hold? Polymath project forum
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Exercise 0.2
aeemc2   4
N Oct 1, 2024 by felixgotti
First, suppose that $n=1$. Note that we can find a polynomial of $\mathbb{N}_0[x]$ that is not irreducible. We can consider, for example, $f(x)=x^3+5x=x(x^2+5)$. Fix $n \in \mathbb{N}_{>1}$ and consider the polynomial $p(x)=x^{n}+1$. Suppose towards a contradiction that $p(x)=\sum_{i=1}^n a_i(x)$ with $a_i(x) \in \mathcal{A}(\mathbb{N}_0[x])$. Note that we have at least one $a_i(x)$ such that $ord(a_i(x)) \neq 0$. Indeed, if $ord(a_i(x))=0$ for every $i \in [[1,n]]$, then the constant coefficient of $p(x)$ is at least $n$. However if $ord(a_i(x) \neq 0$, then we have that $x \mid_{\mathbb{N}_0[x]} a_i(x)$, which is a contradiction with the irreducibility of $a_i(x)$. This proves the result.
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aeemc2
Sep 7, 2024
felixgotti
Oct 1, 2024
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