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MIT PRIMES/Art of Problem Solving

CROWDMATH 2024: Where Does the Goldbach Conjecture Hold?

Problems

The problems in the CrowdMath project are open, unsolved problems in mathematics. We will be discovering new truths that were unknown before.

Each problem has pointers to relevant readings that will help you to get started. Use the "View Discussions" button on a problem to see and reply to all of the comments made so far. We will periodically be updating the summaries of each problem with the progress that we've made.

V t Open Problem 1
The first open problem of this research project is to find all positive real numbers $r$ such that the semidomain $\mathbb{N}_0[r]$ satisfies an analogue of the Goldbach conjecture. You should start by the case when $r$ is rational, so we split the problem into the following two parts.

(a) Find all $q \in \mathbb{Q}^{+}$ such that $\mathbb{N}_0[q]$ satisfies an analogue of the Goldbach conjecture.

(b) Find all $r \in \mathbb{R}^{+}$ such that $\mathbb{N}_0[r]$ satisfies an analogue of the Goldbach conjecture.
V t Open Problem 2
Here are two open problems about Goldbach analogs in connection with localization. Let $\mathfrak{S}$ be a semidomain, and let $S$ be a multiplicative subset of $\mathfrak{S}$.

(a) Assume that every non-constant Laurent polynomial in $S^{-1}\mathfrak{S}[x,x^{-1}]$ can be written as the sum of two irreducibles. Does the same property hold in the semidomain $\mathfrak{S}[x,x^{-1}]$?

(b) Assume now that every nonunit element of $\mathfrak{S}$ can be written as the sum of two irreducibles. Can we write every nonunit element of $S^{-1}\mathfrak{S}$ as the sum of $k$ irreducibles for some $k \in \mathbb{N}$?
a