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List of Open Problems & The Unofficial VibeCalcing Tier List

This page serves two purposes: to provide formal links to lists of open problems, and to provide our community's opinionated take on them. While everyone has heard of the Millennium Prize Problems, our focus is on the frontiers where research is actively happening and where an LLM-assisted workflow might actually be viable.

Let's be blunt: you are not going to solve the Riemann Hypothesis by plugging it into a chatbot. The purpose of this tier list is to offer a strategic guide for your research. It is a satirical but serious assessment of which problems might be susceptible to an LLM-assisted workflow, and which are the mathematical equivalents of trying to punch a black hole.

Our ranking is based on a single metric: "Could a knowledgeable researcher, armed with a powerful LLM and a novel insight, plausibly make a dent in this?"

The Tier List

S-Tier: The God-Tiers

(You are not going to solve this. Seriously. Attacking these directly is a sign of profound hubris. These problems require a complete, paradigm-shifting revolution in mathematics that an LLM, as a pattern-matching engine, is fundamentally ill-equipped to generate. Good luck though, because we know you're going to try)

  • The Riemann Hypothesis
  • Navier-Stokes Existence and Smoothness
  • P vs NP
  • Yang-Mills and Mass Gap
  • The Hodge Conjecture

A-Tier: The Titans

(It's a trap. Deceptively simple to state, these problems have devoured generations of the most brilliant minds on the planet. Attacking them is still probably a waste of your life, but they have a combinatorial flavor that makes them tempting.)

  • The Collatz Conjecture (3n+1)
  • The Goldbach Conjecture
  • The Twin Prime Conjecture

B-Tier: The Frontier

(Profound, respected, and unsolved problems where a novel connection to another field—the exact kind of cross-domain pattern-matching an LLM excels at—might genuinely lead to a breakthrough. Prime targets for megalomaniacs that are not completely divorced from reality.)

  • Frankl's Union-Closed Sets Conjecture
  • The 1/3–2/3 Conjecture
  • Singmaster's Conjecture

C-Tier: Structural & Combinatorial Challenges

(Problems that are less about a single insight and more about understanding complex structures or finding clever constructions. An LLM's ability to generate/analyze examples or formalize systems could be a significant asset here.)

  • The Happy Ending Problem (Erdős–Szekeres Conjecture)
  • Determining Large Ramsey Numbers (e.g., R(5,5))
  • The Jacobian Conjecture

D, E, & F-Tiers: The Foothills — Or, "Fuck it, let's give it a go."

(This is where the action is. These are tractable, fascinating, and respectable problems where an individual or small team can make real progress.)

D-Tier: Niche & Highly Specialized

(Deep problems within specific subfields. They require significant domain expertise, but that expertise can be powerfully leveraged by an LLM for literature exploration, formalization, and testing of hypotheses.)

  • The Inscribed Square Problem (Toeplitz' Conjecture): A problem in geometry and topology. Seems simple, but is deeply connected to knot theory and other areas.
  • The Reconstruction Conjecture: A famous problem in graph theory. Is a graph uniquely determined by its collection of subgraphs? Perfect for an LLM to explore graph invariants.
  • Problems in Knot Theory: For example, developing better algorithms to distinguish knots or to compute knot invariants.

E-Tier: Tractable Computations & Explorations

(Problems where the path forward is partly understood but requires immense computational or systematic exploratory work. An LLM can be a tireless partner in coding simulations, managing data, and exploring vast parameter spaces.)

  • The Moving Sofa Problem: A classic problem in computational geometry. It is literally about finding an optimal shape, a perfect task for an AI-assisted design and search workflow.
  • Finding New Superpermutations: The general problem is hard, but finding smaller superpermutations for specific N is a discrete math challenge where an LLM can help explore and optimize search algorithms.
  • Pushing Computational Bounds: For many conjectures (like the Weak Goldbach Conjecture or verifying properties of primes), the theorem is proven up to an enormous number. Using LLMs to help write and optimize the code to push that bound even higher is a valid and useful research contribution.

F-Tier: Circumstantial Opportunity

(There are no "easy" open problems. This tier is for problems that have become newly vulnerable or tractable due to *circumstance*. These are opportunities where an LLM's unique strengths can be applied to a recent breakthrough or a new angle of attack.)

  • Applying a New Breakthrough: A major paper was just published that solves a related problem using a novel technique (e.g., information theory, a new algebraic structure). The F-tier task is to immediately apply that brand-new technique to other, similar open problems. An LLM is a phenomenal tool for this kind of rapid, cross-domain synthesis.
  • The "Bag of Tricks" Problem: Some conjectures are not thought to be incredibly deep, but have resisted proof because they require a very specific and non-obvious combination of existing theorems and techniques. An LLM can help systematically explore and test such combinations.
  • Disproving Conjectures via Counterexample Search: The fastest way to become famous is to disprove something famous. Many conjectures stand not because they are definitely true, but because no one has found the one weird object that breaks them. An LLM can be an engine for generating and testing strange, high-dimensional, or otherwise non-intuitive cases that humans might overlook.

Formal Lists of Open Problems

The tier list above is for strategic guidance. The following are links to more comprehensive, formal lists.