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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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Where Does the Goldbach Conjecture Hold? Polymath project forum
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exercise 1.2
aeemc2   4
N Nov 10, 2024 by felixgotti
Hi professor. I wanted to ask you if the correct statement for exercise 1.2 is "Let $S$ be an additively reduced semidomain containing an additive atom that is not a multiplicative unit. Prove that no analog of the Goldbach conjecture holds for $S[x]$" instead of $S$. I think I have the answer if it is for $S[x]$. It is enough to consider the polynomial $f(x)=ax^n+ax^{n-1}+ax^{n-2}+...+ax+a$ that cannot be expressed as the sum of a finite number of irreducibles.
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aeemc2
Nov 1, 2024
felixgotti
Nov 10, 2024
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