Goldbach's Conjecture

Prove Goldbach's conjecture by showing that every even integer greater than 2 can be expressed as the sum of two prime numbers, thereby establishing the strong (binary) form of the conjecture.

Background

In the Recommendations section, the author argues that mathematicians should embrace AI to advance core mathematical goals. As an illustrative example of high-impact objectives, the essay explicitly mentions proving Goldbach's conjecture alongside realizing the Langlands program and resolving the P ≠ NP problem.

Goldbach's conjecture is a longstanding open problem in number theory asserting that every even integer greater than 2 is the sum of two primes. The essay invokes this conjecture to exemplify the kind of major theorem-proving milestones that would underscore AI's positive role in mathematical discovery.

References

If AI can help us realize the Langlands program, prove Goldbach's conjecture, and resolve the $P \ne NP$ problem, will that be all that bad?

— Mathematicians in the age of AI  (2603.03684 - Avigad, 4 Mar 2026) in Section 4 (Recommendations)

This conjecture is still wide open, but has been computationally verified for integers $\leq 4\cdot 10{18}$ (see ).

— Computational results on sums of a prime with squares or cubes  (2609.20505 - Applegate et al., 17 Sep 2026) in Section 1, paragraph 2

The famous Goldbach conjecture remains open for nearly three centuries.

— Prime Multiple Missing Graphs  (2501.02529 - Ghosh, 5 Jan 2025) in Section 1, Introduction

But the major problem is to show analytically that the exact value of $T(n)=\frac{f(n)}{n}<1$ or to be precise less than $1-\frac{2}{n}$, where $1-T(n)$ is the probability of a vertex (in the desired set) to be a neighbor of $x$.

— Counting degrees of vertices in near Goldbach graphs  (2608.14159 - Ghosh et al., 14 Aug 2026) in Conclusion, final section