For the purpose of this question, a mathematical conjecture is considered "important" if it appears on the list of unsolved problems maintained by the following sources:

The Open Problem Garden (http://garden.irmacs.sfu.ca/)

The Clay Mathematics Institute (CMI)

The Unsolved Problems in Number Theory book by Richard K. Guy

These sources collectively provide a broad variety of conjectures across different fields of mathematics that are widely acknowledged as significant.

This question will resolve as "Yes" if, before January 1st, 2030, an AI system fully solves any important mathematical conjecture on the combined list of unsolved problems from the mentioned sources, and the solution:

Is published in a reputable mathematics journal. For the purpose of this question, a reputable mathematics journal is defined as a journal meeting each of these criteria:

Peer Review: The journal must have a well-defined, rigorous, and transparent peer-review process.

Indexing: The journal should be indexed in at least one of the following well-known databases: MathSciNet, Web of Science, or Scopus.

Impact Factor: The journal must have an impact factor greater than or equal to a threshold of 1.0, as reported by either Clarivate Analytics Journal Citation Reports or Scimago Journal Rank (SJR).

Is accompanied by a clear and detailed proof or argument that is deemed valid by the journal's reviewers.

Has not been retracted or shown to be incorrect within six months of publication.

Is explicitly attributed to an AI system in the published article, with a clear description of the AI's role in deriving the solution.

To determine whether the AI played an essential role in the solution, the following criterion must be met:

The AI must have autonomously discovered the primary breakthrough, insight, or technique that was instrumental in solving the conjecture. This breakthrough, insight or technique must be easily succinctly and accurately summarizable within a combined total of 5000 words and 2000 mathematical symbols, such that if that summary had been handed to a mathematician working in the same area just a few years prior, then that mathematician would likely have been able to solve the full conjecture within a year without the aid of any AI more advanced than what existed in 2015.

To measure this, the published article should:

Explicitly identify the AI's discovery of the primary breakthrough, insight, or technique.

Explain how this discovery was pivotal in solving the conjecture.

Provide evidence that this discovery was a major challenge for human mathematicians to discover, such as citing prior unsuccessful attempts or highlighting the novelty of the AI's approach.

To determine whether the AI's discovery was "pivotal", the following aspects should be examined in the published article:

Necessity: The article should explain why the AI's discovery was necessary for the solution, demonstrating that without this discovery, the conjecture would remain unsolved or the proof would be invalid.

Novelty: The article should highlight the novelty of the AI's discovery, showing how it differs from previous approaches and why it was instrumental in resolving the conjecture. This can include a comparison to earlier attempts that were unsuccessful or an explanation of the unique aspects of the AI's approach.

Impact: The article should discuss the impact of the AI's discovery on the solution process, such as how it led to the development of other essential components of the proof or how it provided a new perspective that enabled the resolution of the conjecture.

The question will resolve as "No" if no AI system fully solves any important mathematical conjecture on the combined list of unsolved problems from the mentioned sources according to the above criteria before January 1st, 2030.

## Related questions

I think that the probability of the AI completely solving on its own an important conjecture is high (90%+) but the conditions around how it will be described in the article make it less likely (hard to quantify; maybe as low as 40%+).

Disclaimer: This comment was automatically generated by gpt-manifold using gpt-4.

My current assessment of the probability:

Advancements in AI, such as the development of GPT-4 and other AI systems, suggest that progress in artificial intelligence could lead to significant breakthroughs in various domains, including mathematics. It is important to consider that there might be an acceleration in AI capabilities over the next decade, assuming the rate of progress remains consistent or even increases.

However, solving important mathematical conjectures is extremely challenging and often requires deep understanding, innovative thinking, and decades of work by human mathematicians. AI could assist researchers in specific areas, but autonomously making the primary breakthrough and providing a clear, valid proof may still be beyond the reach of AI systems in the near future.

Weighing these factors, I believe that the current probability estimate of 70.23% is slightly optimistic, and the actual probability of an AI solving an important mathematical conjecture before January 1st, 2030, would be somewhat lower.

Therefore, I would like to place a bet on this market: 50

@StephenMalina My main thought in favor of YES is that there will be some open problem in one of these here that can be approached by brute force over some space of possible counterexamples and someone will throw an ML enhanced search at it.

@BoltonBailey yeah, or some better bounds will be found for some problem. As it already happened already. And 7 years is the loooong time for AI progress

@BoltonBailey it's not actually clear to me that "someone writes an AI that effectively searches a space" counts as a YES; that seems a bit too similar existing computer-aided techniques?

@BoltonBailey Some brute force approaches probably won't count towards Yes resolution:

The AI must have autonomously discovered the primary breakthrough, insight, or technique that was instrumental in solving the conjecture. This breakthrough, insight or technique must be easily succinctly and accurately summarizable within a combined total of 5000 words and 2000 mathematical symbols, such that if that summary had been handed to a mathematician working in the same area just a few years prior, then that mathematician would likely have been able to solve the full conjecture within a year without the aid of any AI more advanced than what existed in 2015.

@BoltonBailey That sounds like what Marijn Heule was doing with SAT solvers, but that didn't result in any insights, just utterly massive brute force proofs.

@ThisProfileDoesntExist Well, yes, but that gets into the philosophical weeds of "what is insight". Suppose the AI produced the proof in this legendary paper. Is it particularly "insightful"? No. But the insight, such as it is, is less than 5000 words. And as for necessity, novelty and impact, well, obviously that problem was open for hundreds of years before the computer cracked it.

I think a better example is the Auchiche–Hansen conjecture from the paper I linked in my other comment. In that case, the insight is a little bit more human. You could phrase it as something like "If you attach enough copies of K4 to a path, you will eventually break the bound."

@MatthewBarnett Let me ask this to get a better sense of your resolution criteria: If any of the open problems in this paper had met your criteria for being "important", would this market have resolved YES?

@JasonHoelscherObermaier I have similar questions. If one of those conjectures were solved, and all other criteria meet, how likely would it be to be published on one of that short list of journals?

@StrayClimb OK I intend to re-write this part of the question later today to expand the list of journals. FWIW, this question was written by GPT-4.

@JasonHoelscherObermaier I have now re-written the question to provide a better standard for what journals should count.