This page collects the various ways in which AI tools have contributed to the understanding of ErdΕs problems. Note that a single problem may appear multiple times in these lists. For further discussion of these contributions, see this forum.
Legend
- π’ : full solution to the problem
- π‘ : partial solution to the problem
- π΄ : failure to make progress on the problem
Disclaimers
One should interpret the contributions listed here with the following caveats in mind.
- ErdΕs problems vary widely in difficulty (by several orders of magnitude), with a core of very interesting, but extremely difficult problems at one end of the spectrum, and a "long tail" of under-explored problems at the other, many of which are "low hanging fruit" that are very suitable for being attacked by current AI tools. Unfortunately, it is hard to tell in advance which category a given problem falls into, short of an expert literature review. (However, if an ErdΕs problem is only stated once in the literature, and there is scant record of any followup work on the problem, this suggests that the problem may be of the second category.) In particular, I would advise against direct comparison of raw counts of problems solved by one methodology or another, as this may not be an apples-to-apples comparison. In particular, one should not view this page as representing any sort of finely calibrated benchmark.
- Many of the problems on the site currently lack a thorough literature review, and in the absence of such a review, any designation of a problem as "open" should be viewed as provisional. In particular, it has already happened multiple times that an AI tool has managed to solve a problem listed as open on the site, only to find shortly afterwards that the problem was in fact solved in the literature (though sometimes by a slightly different method); see Section 2 below.
- The reporting of these AI tools on this page is highly incomplete, particularly with regards to negative results. One should keep selection bias in mind before drawing any conclusions about the success rate of these tools.
- In a few cases ErdΕs stated the problem incorrectly (or the problem was not stated properly on the web site), allowing for the problem as literally stated to be solved on a technicality, even though this was not the actual question ErdΕs intended to ask. Working out what ErdΕs actually intended in such cases requires a somewhat subjective contextual analysis guided by expertise in the subject.
- If an ErdΕs problem was first posed
$N$ years ago, and was recently solved by an AI tool, it may be misleading to make announcements such as "this problem resisted all human efforts at proof for$N$ years" in order to imply that the problem is particularly difficult. Such an announcement may be technically true, especially if a comprehensive literature review turns up no significant prior literature on the problem, but it is also possible that this particular question received very little attention from human mathematicians during this time period, and the absence of any past progress may therefore be more a reflection of the obscurity of the problem than of its difficulty. (If on the other hand there is substantial literature expending significant effort on establishing partial results, without generating a clear path to a full solution, this is more convincing evidence of difficulty than mere absence of literature.) - The mathematical interest in these problems does not stem purely from their solution, but also on what insights and lessons the work on these problems sheds on related topics, and how the problem fits into the broader context of the field. When a solution is human-generated, such additional connections to the rest of the subject tend to be provided organically through the remarks and comments generated in the writing process (which often includes a literature review step). However, when a solution is primarily AI-generated, this useful penumbra of additional context may be largely absent, rendering the final product of less mathematical utility even if it does technically solve the stated problem. Thus, contributions to ErdΕs problems should be evaluated holistically, taking into account the extent to which the results have been integrated into the existing web of knowledge around the problem, in addition to whether the problem itself was solved.
- For the above reasons, not every solution to an open ErdΕs problem would automatically qualify as a publishable paper in a reputable math journal, especially if the problem has not attracted significant prior literature, and the proof techniques ended up being minor modifications of existing methods. However, even when the problem solved was obscure and the proof was routine, it is still worth having the solution recorded somewhere in the published literature; the usual solution in such situations is to bundle such solutions as components of a larger paper which makes additional contributions such as a broader literature review, an exploration of the strengths and weaknesses of known methods (using such solutions as examples), and highlighting the open problems that continue to lie out of reach of these methods.
- For additional guarantees of correctness, it is considered good practice to formalize AI-generated proofs in a proof assistant language such as Lean. However, there are still "exploits" available in such guarantees, for instance if the formal proof introduces additional axioms, or if there is a misformalization of the problem statement that may exploit some quirk of the math library or syntax used in the formalization. An AI-generated formal proof that is suspiciously short and elementary (or suspiciously verbose and trivial) is particularly subject to this issue, particularly in the hands of a user without extensive experience formalizing proofs by hand.
I just saw an exciting announcement on social media about AI being used to solve an ErdΕs problem. Why can't I find out about it on this page?
If the social media post is very recent, then most likely the claim is undergoing an expedited, but not instantaneous, peer review over at the ErdΕs problem web site. The discussion page for the ErdΕs problem in question is most likely the best place to get the most recent updates, and in particular to perform the due diligence needed to take into account the caveats listed above; until then, the problem may be placed in the "pending assessment" category below. Once the discussion on that page has reached consensus, this contribution will most likely be moved from "pending assessment" to a different category on this site; but until then, I recommend taking any bold claims about this achievement with a grain of salt, and check back in a couple hours or days for a more accurate update.
1. AI-generated solutions, partial solutions, or negative results for previously open problems
| Problem | AI tools used | Date | Outcome |
|---|---|---|---|
| [36] | AlphaEvolve | 3 Nov, 2025 | π‘ Slight improvement to past construction |
| [51] | ChatGPT (free version) | 11 Jan, 2026 | π΄ Incorrect proof found |
| [52] | AlphaEvolve | 3 Nov, 2025 | π΄ Did not match past constructions |
| [64] | AlphaEvolve | 3 Nov, 2025 | No counterexample found |
| [67] | AlphaEvolve | 3 Nov, 2025 | π΄ Did not match past constructions |
| [106] | AlphaEvolve | 3 Nov, 2025 | Matched past construction |
| [124] | Aristotle | 29 Nov, 2025 | π‘ Partial result (Lean) |
| [205] | Aristotle, ChatGPT 5.2 Thinking | 10 Jan, 2026 | π’ Full solution (Lean) |
| [391] | AlphaEvolve | 3 Nov, 2025 | π΄ Did not match past constructions |
| [477] | AlphaProof | 7 Jan, 2026 | π‘ Proof of variant problem (Lean) |
| [486] | ChatGPT 5.2 | 11 Jan, 2026 | π‘ Cheap counterexample to previous formulation of the problem |
| [493] | AlphaEvolve | 3 Nov, 2025 | No counterexample found |
| [507] | AlphaEvolve | 3 Nov, 2025 | π‘ Surpassed some past constructions |
| [728] | Aristotle, ChatGPT 5.2 Pro | 6 Jan, 2026 | π’ Full solution (Lean), using arguments similar to π‘ Pomerance (2014) |
| [729] | Aristotle, ChatGPT 5.2 Pro | 8-10 Jan, 2026 | π’ Full solution (Lean), using a modification of the solution to #728 |
| [949] | AlphaProof | 7 Jan, 2026 | π‘ Proof of variant problem (Lean) |
| [1097] | AlphaEvolve | 3 Nov, 2025 | π‘ Slight improvement to past construction |
2. Fully AI-generated solutions to problems thought open, but for which an earlier human solution was subsequently found
| Problem | AI tools used | Date | Outcome | Literature result | Literature result found on | Similar proofs? |
|---|---|---|---|---|---|---|
| [333] | ChatGPT 5.2 Pro, Claude Opus 4.5 | 25 Dec, 2025 | π’ Full solution (Lean) | π’ ErdΕs and Newman (1977) | 25 Dec, 2025 | Yes |
| [397] | Aristotle, ChatGPT 5.2 Pro | 10 Jan, 2026 | π’ Full solution (Lean) | π‘ Elkies (2013) [likely upgradeable to full solution] | 10 Jan, 2026 | No |
| [659] | Gemini 3.0 | 13 Jan, 2026 | π’ Full solution (Lean) | π‘ Moree and Osburn (2006), π‘ Sheffer (2014) [upgradeable to full solution] | 13 Jan, 2026 | Yes |
| [897] | Aristotle, Archivara | 26 Dec, 2025 | π’ Full solution (Lean) | π’ Wirsing (1981) | 26 Dec, 2025 | Yes |
| [1026] | Aristotle | 7 Dec, 2025 | π’ Full solution (Lean) | π’ Tidor, Wang, and Yang (2016) | 8 Dec, 2025 | Only after applying an argument from Seidenberg (1959) |
| [1077] | Aristotle | 24 Dec, 2025 | π‘ Trivial counterexample to previous formulation of problem (Lean) | π’ Jiang and Longbrake (2025) | 28 Dec, 2025 | No |
3. AI-generated tools applied to problems previously known to be solved (or partially solved)
| Problem | AI tools used | Date | Literature result | Outcome |
|---|---|---|---|---|
| [43] | Aristotle | 4 Dec, 2025 | π‘ Barreto (2025) | π‘ New proof of partial result (Lean) |
| [124] | Gemini DeepThink | 30 Nov, 2025 | π‘ Aristotle (2025) | π΄ Did not reproduce a known partial result despite hints |
| [198] | AlphaProof | 2025 | π’ Baumgartner (1975) | π’ New proof found |
| [264] | Aristotle | 18 Dec, 2025 | π‘ KovaΔ and Tao (2024) | π‘ New proof of partial result (Lean); π΄ no progress on remaining open problems |
| [379] | Seed-Prover 1.5 | 21 Dec, 2025 | π’ Cambie, KovaΔ and Tao (2025) | π’ New proof found |
| [392] | GPT-5.2 Pro, Aristotle | 31 Dec, 2025 | π’ ErdΕs and Graham (unpublished, 1980), Cambie (2025) | Unable to reconstruct proof |
| [488] | Aristotle | 27 Nov, 2025 | π‘ Cambie (2025) | π‘ New disproof of alternate version (Lean) |
| [493] | SeedProver, Aristotle, ChatGPT | 2025 | π’ Seamans (2025) | π’ New or existing proof found (Lean) |
| [679] | Aristotle | 12 Jan, 2026 | π‘ DottedCalculator (2025) | π‘ Improved proof by reducing dependence on prime number theorem |
| [871] | Claude Opus 4.5, Gemini 3 Pro | 5 Jan, 2026 | π‘ ErdΕs and Nathanson (1989) | π’ Existing partial result upgraded to full solution (Lean) |
| [942] | Gemini | 23 Nov, 2025 | π‘ ErdΕs (unpublished, 1976) | π‘ New proof found |
| [958] | Aristotle | 27 Dec, 2025 | π’ Clemen, Dumitrescu, and Liu (2025) | π’ New proof found (Lean) |
| [1043] | Aristotle | 28 Dec, 2025 | π’ Pommerenke (1961) | π’ New proof found (Lean) |
| [1043] | ChatGPT | 30 Dec, 2025 | π’ Pommerenke (1961), Aristotle | π’ New proof found |
| [1095] | Aristotle | 30 Dec, 2025 | π‘ Ecklund, ErdΕs, and Selfridge (1975) | π‘ New proof of (slightly weaker) partial result found (Lean); π΄ no progress on remaining open problems |
4. Solutions generated by humans in collaboration with AI
| Problem | Human collaborators | AI tools used | Date | Outcome |
|---|---|---|---|---|
| [367] | Boris Alexeev, Wouter van Doorn, Terence Tao | Gemini Deepthink, Aristotle | 20-22 Nov, 2025 | π‘ Partial result |
| [401] | Boris Alexeev, Kevin Barreto, Leeham, Nat Sothanaphan | Aristotle, ChatGPT-5.2 Pro | 10-11 Jan, 2026 | π’ Counterexample to alternate formulation; full solution to revised formulation (Lean) |
| [848] | Mehtaab Sawhney, Mark Sellke | GPT-5 | 12 Oct-20 Nov, 2025 | π’ Full solution |
| [1026] | Boris Alexeev, Stijn Cambie, Terence Tao, Lawrence Wu | ChatGPT, Aristotle, AlphaEvolve, Gemini | 8 Dec, 2025 | π’ Full solution (Lean) |
| [1038] | Junnosuke Koizumi, jspier, Nat Sothanaphan, Terence Tao | ChatGPT, AlphaEvolve | In progress | π‘ Partial result |
5. Problems with an AI-powered literature review
| Problem | AI tools used | Review performed on | Outcome |
|---|---|---|---|
| [35] | GPT-5 | 13 Oct, 2025 | π‘ Partial results found |
| [66] | GPT-5 | 13 Oct, 2025 | π‘ Partial results found |
| [94] | GPT-5 | 2 Nov, 2025 | π’ Full solution found |
| [124] | ChatGPT DeepResearch, Gemini DeepResearch | 30 Nov, 2025 | No significant results found |
| [167] | GPT-5 | 12 Oct, 2025 | π‘ Partial results found |
| [188] | GPT-5 | 13 Oct, 2025 | π‘ Partial results found |
| [203] | ChatGPT DeepResearch, Gemini DeepResearch | 19 Oct, 2025 | No significant results found |
| [205] | ChatGPT DeepResearch | 10 Jan, 2026 | No significant results found |
| [223] | GPT-5 | 13 Oct, 2025 | π’ Full solution found |
| [248] | Gemini DeepResearch | 19 Oct, 2025 | No significant results found |
| [330] | ChatGPT DeepResearch, Gemini DeepResearch, Claude | 19 Dec, 2025 | π΄ Partial results found (with inaccuracies); did not find literature proof |
| [333] | ChatGPT 5.2 Pro | 25 Dec, 2025 | π΄ Incorrect proof claimed; did not find literature proof |
| [334] | Gemini DeepResearch | 19 Nov, 2025 | No significant results found |
| [339] | GPT-5 | 11 Oct, 2025 | π’ Full solution found |
| [347] | ChatGPT DeepResearch | 25 Oct, 2025 | π’ Full solution found |
| [354] | ChatGPT DeepResearch | 19 Oct, 2025 | π‘ Partial results found |
| [367] | ChatGPT DeepResearch, Gemini DeepResearch | 22 Nov, 2025 | π΄ No significant results found; did not find ErdΕs problems community proof |
| [370] | ChatGPT DeepReseach, Gemini, Gemini DeepResearch | 17 Oct, 2025 | π‘ Problem found to be misstated; solutions to variants found |
| [387] | ChatGPT DeepResearch | 1 Nov, 2025 | π‘ Partial results found |
| [397] | ChatGPT DeepResearch | 10 Jan, 2026 | π’ Full solution found |
| [401] | Claude | Jan 10, 2026 | No significant results found |
| [421] | ChatGPT DeepResearch, Gemini DeepResearch | 18 Oct, 2025 | No significant results found |
| [434] | GPT-5 | 29 Oct, 2025 | π΄ Literature proof not found |
| [481] | ChatGPT DeepResearch, Gemini DeepResearch | 1 Dec, 2025 | π‘ Unwittingly reproduced existing proof |
| [481] | ChatGPT | 3 Dec, 2025 | π’ Full solution found |
| [494] | GPT-5 | 13 Oct, 2025 | π’ Full solution found |
| [515] | GPT-5 | 15 Oct, 2025 | π’ Full solution found |
| [516] | ChatGPT DeepResearch, Claude, Gemini DeepResearch | 28 Dec, 2025 | π’ Problem found to be misstated; full solution to actual problem found |
| [524] | ChatGPT 5.2 Pro | 27 Dec, 2025 | π‘ Partial results found |
| [559] | ChatGPT DeepResearch, Gemini DeepResearch | 26 Oct, 2025 | π‘ Partial results found |
| [621] | GPT-5 | 13 Oct, 2025 | π’ Full solution found |
| [645] | GPT-5 | 20 Oct, 2025 | π’ Full solution found |
| [659] | Gemini DeepResearch | 13 Jan, 2026 | π‘ Partial solution (upgradeable to full solution) found |
| [686] | ChatGPT, Gemini DeepThink, Gemini DeepResearch | 18 Oct, 2025 | No significant results found |
| [689] | ChatGPT DeepResearch, Gemini DeepResearch | 29 Oct, 2025 | No significant results found |
| [672] | ChatGPT | 7 Jan, 2025 | π‘ Partial results found |
| [700] | ChatGPT DeepResearch | 17 Dec, 2025 | No significant results found |
| [707] | ChatGPT | 22 Oct, 2025 | π΄ Literature proof not found |
| [728] | ChatGPT DeepResearch | 15 Dec, 2025 | No significant results found |
| [729] | ChatGPT DeepResearch | Jan 10, 2026 | π‘ Related results found |
| [737] | GPT-5 | 30 Sep, 2025 | π’ Full solution found |
| [750] | GPT-5 | 13 Oct, 2025 | π‘ Partial results found |
| [786] | ChatGPT DeepResearch, Gemini DeepResearch | 18 Oct, 2025 | No significant results found |
| [788] | GPT-5 | 14 Oct, 2025 | π‘ Partial results found |
| [793] | ChatGPT DeepResearch | 30 Nov, 2025 | π‘ Partial results found |
| [811] | GPT-5 | 13 Oct, 2025 | π‘ Partial results found |
| [822] | GPT-5 | 13 Oct, 2025 | π’ Full solution found |
| [827] | GPT-5 | 14 Oct, 2025 | π‘ Partial results found |
| [829] | GPT-5 | 13 Oct, 2025 | π‘ Partial results found |
| [871] | ChatGPT DeepResearch | 6 Dec, 2025 | π‘ Partial results found |
| [903] | GPT-5 | 14 Oct, 2025 | π’ Full solution found |
| [906] | ChatGPT DeepReseach, Gemini DeepResearch | 17 Oct, 2025 | No significant results found |
| [915] | ChatGPT DeepResearch | 26 Oct, 2025 | π‘ Partial results found |
| [940] | ChatGPT DeepResearch | 24 Oct, 2025 | No significant results found |
| [942] | ChatGPT DeepResearch | 23 Nov, 2025 | π‘ Partial results found |
| [965] | ChatGPT | 2 Jan, 2026 | π’ Full solution found |
| [967] | GPT-5 | 29 Oct, 2025 | No significant results found |
| [990] | ChatGPT DeepResearch, Gemini DeepResearch | 20 Oct, 2025 | π‘ Partial results found |
| [1002] | ChatGPT DeepResearch, Gemini DeepResearch | 3 Nov, 2025 | No significant results found |
| [1008] | GPT-5 | 29 Sep, 2025 | π’ Full solution found |
| [1011] | GPT-5 | 13 Oct, 2025 | π‘ Partial results found |
| [1016] | ChatGPT DeepResearch, Gemini DeepResearch | 18 Oct, 2025 | π‘ Partial results found |
| [1019] | ChatGPT DeepResearch, Gemini DeepResearch | 18 Oct, 2025 | π‘ Solution claim found, but precise citation not found |
| [1022] | ChatGPT DeepResearch, Gemini DeepResearch | 4 Dec, 2025 | No significant results found |
| [1038] | Gemini DeepResearch | 16 Nov, 2025 | No significant results found |
| [1041] | GPT-5 | 16 Nov, 2025 | No significant results found |
| [1043] | GPT-5 | 12 Oct, 2025 | π’ Full solution found |
| [1079] | GPT-5 | 13 Oct, 2025 | π’ Full solution found |
| [1099] | ChatGPT DeepResearch, Gemini, Gemini DeepResearch | 19 Oct, 2025 | π’ Full solution found |
| [1124] | ChatGPT | 31 Dec, 2025 | π’ Full solution to proposed variant found |
6. Proofs that were formalized by AI
| Problem | Proof to be formalized by | AI tools used to formalize | Formalized on | Other known proofs |
|---|---|---|---|---|
| [26] | π’ Ruzsa | Aristotle | 28 Dec, 2025 | |
| [31] | π’ Lorentz (1954), Wouter van Doorn (2025) | ChatGPT, Aristotle | 24 Nov, 2025 | |
| [43] | π‘ Barreto (2025) | Claude, Aristotle | 21 Dec, 2025 | |
| [56] | π’ Ahlswede-Khachatrian (1995) | ChatGPT, Aristotle | 25 Nov, 2025 | |
| [105] | π’ Xichuan (2025) | ChatGPT Pro, Aristotle | 17 Nov, 2025 | |
| [106] | π‘ Baek, Koizumi, and Ueoro (2024) | Aristotle | 10 Dec, 2025 | |
| [189] | π’ KovaΔ (2023) | Gemini 3 Pro, Aristotle | 17 Dec, 2025 | |
| [198] | π’ Baumgartner (1975) | ChatGPT, Aristotle | 24 Nov, 2025 | |
| [226] | π’ Sato and Rankin (1974) | Aristotle | 29 Dec, 2025 | π’ Barth and Schneider (1970) |
| [229] | π’ Barth and Schneider (1972) | Aristotle | 28 Dec, 2025 | |
| [246] | π’ Birch (1959) | Aristotle | 28 Dec, 2025 | |
| [303] | π’ Brown and RΓΆdl (1991) | SeedProver | 21 Dec, 2025 | |
| [337] | π’ Ruzsa and TurjΓ‘nyi (1985) | Aristotle | 10 Dec, 2025 | |
| [350] | π’ Ryavec (1974) | ChatGPT, Aristotle | 25 Nov, 2025 | |
| [367] | π‘ ErdΕs problems community (2025) | Aristotle | 22 Nov, 2025 | |
| [370] | π’ Steinerberger (2025) | ChatGPT, Aristotle | 24 Nov, 2025 | |
| [418] | π’ Browkin and Schinzel (1995) | ChatGPT, Aristotle | 22 Nov, 2025 | |
| [480] | π’ Chung and Graham (1984) | Aristotle | 28 Nov, 2025 | |
| [481] | π’ Barreto (2025) | Claude, Aristotle | 1 Dec, 2025 | π’ Klarner (1982) |
| [499] | π’ Marcus and Minc (1962) | Aristotle | 29 Nov, 2025 | |
| [541] | π’ Grynkiewicz (2011) | ChatGPT, Aristotle | 30 Dec, 2025 | π’ Gao, Hamidoune, and Wang (2010) |
| [613] | π’ Pikhurko (2001) | ChatGPT Pro | 4 Nov, 2025 | |
| [645] | π’ Brown and Landman (1999) | ChatGPT, Aristotle | 23 Nov, 2025 | |
| [678] | π’ Cambie (2025) | Aristotle | 7 Jan, 2026 | |
| [707] | π’ Hall (1947) | ChatGPT | 23 Nov, 2025 | π’ Alexeev and Mixon (2025) |
| [845] | π’ van Doorn and Everts (2025) | Aristotle | 8 Jan, 2026 | |
| [897] | π’ ErdΕs and Wirsing (1975) | Aristotle | 27 Nov, 2025 | |
| [958] | π’ Clemen, Dumitrescu, and Liu (2025) | Bytedance Seed AI4Math | 2025 | |
| [967] | π’ Yip (2025) | Aristotle | 19 Dec, 2025 | |
| [1000] | π’ Haight | ChatGPT, Aristotle | 28 Dec, 2025 | |
| [1034] | π’ Ma-Tang (2025) | Aristotle | 4 Dec, 2025 | |
| [1080] | π’ De Caen and SzΓ©kely (1992) | Aristotle | 28 Dec, 2025 |
7. Human solutions with some secondary AI tool use
| Problem | Result | Date | AI tools used | Nature of tool use |
|---|---|---|---|---|
| [69], [248], [946] | π’π’π‘ Tao and Teravainen | 1 Dec, 2025 | ChatGPT | Numerics and image generation |
| [114] | π‘ Tao | 13 Dec, 2025 | Gemini, AlphaEvolve | Initial numerics, code and image generation |
| [682] | π’ Gafni and Tao | 8 Aug, 2025 | ChatGPT | Initial code generation |
8. AI tools used to rewrite an existing argument
| Problem | AI tools used | Date | Previous argument |
|---|---|---|---|
| [728] | Aristotle, ChatGPT | 5-7 Jan, 2026 | π’ Barreto / ChatGPT / Aristotle (2026) |
9. AI tools used to perform numerical exploration
| Problem | Date | AI tools used | Nature of tool use |
|---|---|---|---|
| [271] | 11 Jan, 2026 | Unspecified | OEIS sequence generation |
| [872] | 11 Jan, 2026 | ChatGPT | OEIS sequence generation |
10. Pending assessment
| Problem | AI tools used | Date | Status |
|---|---|---|---|
| [None currently] |