Selected mathematical problems
The frontiers
of mathematics.
Updated
Track the best known bounds and unresolved cases for six open problems.
Compare published results with recent claims, and follow each milestone back to its source.
Explore the problemsChoose a question
01 — 06 / Selected historiesThe milestones
Select a resultPreprints are hidden. Full RH is shown only as a conjectured reference.
Excluded under this statement Permitted region Critical line β = ½
A zero is written β + iγ: β is its horizontal position and γ its height. The restriction applies at every height. Shading shows permitted locations, not a distribution of zeros.
Share on the line
Critical-line source ↗Location of every zero
7/8 is stronger than 11/12 because it is closer to ½. These controls compare strength, not publication order: the 7/8 preprint is dated September 30, 2026; the simpler 11/12 argument appeared on October 5. Neither is full RH.
Methodology
How to read
the charts
01
What the numbers measure
Each chart tracks a specific bound or set of cases. These quantities cannot be combined into a “percentage solved” or used to predict when a proof will arrive.
02
Dates and step changes
Lines step when a recorded bound changes. These are selected milestones, not complete record histories. Dates follow the cited announcement or paper; publication dates are noted separately.
03
Publication status
In the timelines, green marks published results or established constructions; amber marks preprints and reports. The region diagram uses shading for permitted and excluded locations, with evidence status shown separately. Full RH is an unproved reference. Status reflects the review date, even when a point is dated to an earlier preprint. The sources document the claims; this site does not independently verify the proofs.
Data and sources
Download data ↓Curated and source-checked October 20, 2026. Research and implementation are AI-assisted; this is an editorial collection, not a mathematical peer review. AI annotations describe documented contributions to a specific result. An absent annotation means no contribution is recorded here.
Twin prime gaps · 5 milestones
| Date | Value | Evidence & attribution | Context & source |
|---|---|---|---|
| < 70,000,000 | Published resultYitang Zhang | Zhang proved that infinitely many consecutive prime pairs have a gap below 70 million. Accepted in May 2013; published in Annals of Mathematics in 2014. Primary source ↗ | |
| ≤ 4,680 | Published resultD. H. J. Polymath / Polymath8a | A worldwide collaboration rapidly reduced the bound. Tao’s September 2013 account records 4,680; the paper followed in 2014. Primary source ↗ | |
| ≤ 600 | Published resultJames Maynard; method independently developed by Terence Tao | Maynard’s preprint gives a bound of 600 with a different sieve method. Published in Annals of Mathematics in 2015. Primary source ↗ | |
| ≤ 246 | Published resultD. H. J. Polymath / Polymath8b | The collaboration refined the new method and proved the unconditional bound 246. Published in Research in the Mathematical Sciences in 2014. Primary source ↗ | |
| ≤ 186 | PreprintOpenAI / GPT-6 Astra | The manuscript claims an unconditional mathematical bound of 186. Its Lean formalization assumes three input axioms, covering exponential-sum estimates and numerical bounds. The repository makes these dependencies explicit. Primary source ↗ · Supporting source ↗AI contribution: The paper credits GPT-6 Astra with the proof. The Lean development remains conditional on input axioms; numerical checks use Python/FLINT. |
Riemann hypothesis · 5 milestones · 3 region comparisons
| Date | Value | Evidence & attribution | Context & source |
|---|---|---|---|
| ≥ 34.74% | Published resultNorman Levinson | Levinson proved that more than one third of the zeros lie on the critical line. The numerical 34.74% version is documented in the Bui–Conrey–Young paper’s historical introduction. Primary source ↗ · Supporting source ↗ | |
| > 40.88% | Published resultJ. Brian Conrey | Conrey’s published result increased the lower bound to more than 40.88%. The numerical value is documented in the Pratt–Robles–Zaharescu–Zeindler introduction. Primary source ↗ · Supporting source ↗ | |
| ≥ 41.05% | Published resultHung Bui, Brian Conrey, Matthew Young | Theorem 1.1 establishes a lower bound of 41.05%. First posted in 2010 and published in Acta Arithmetica in 2011. Primary source ↗ | |
| > 41.7293% | Published resultKyle Pratt, Nicolas Robles, Alexandru Zaharescu, Dirk Zeindler | The paper’s refined numerical bound is 41.7293%, slightly better than five twelfths. The preprint appeared in 2018; publication followed in 2020. Primary source ↗ | |
| ≥ 67.25% | PreprintLevent Alpöge and Ralph Furman | The preprint proves a lower bound of 67.25% for zeros that are both simple and on the critical line. The authors say they verified the argument and take responsibility for it. Primary source ↗AI contribution: The authors credit Claude with discovering and writing the argument. They checked it themselves; journal peer review is not established here. |
Zero-free region comparisons
Ordered by strength, not publication date. These bounds describe horizontal position, not a proportion of zeros. Full RH is an unproved reference.
| Date | Permitted location | Evidence & attribution | Context & source |
|---|---|---|---|
| 1/12 ≤ β ≤ 11/12 | PreprintOpenAI | The preprint states that ζ(s) has no zeros with real part greater than 11/12. Symmetry then confines all nontrivial zeros to 1/12 ≤ β ≤ 11/12. This paper isolates a simpler argument for a weaker bound than 7/8. Source ↗AI contribution: OpenAI is the listed author of the preprint. | |
| 1/8 ≤ β ≤ 7/8 | PreprintOpenAI | The preprint states that ζ(s) has no zeros with real part greater than 7/8. Symmetry then confines all nontrivial zeros to 1/8 ≤ β ≤ 7/8. This narrower strip still permits zeros away from the critical line. Source ↗AI contribution: OpenAI is the listed author of the preprint. | |
| β = ½ | Conjecture · unprovedBernhard Riemann · overview by the Clay Mathematics Institute | Full RH requires every nontrivial zero to have real part exactly ½, without exceptions. The permitted strip would collapse to the critical line. This is a conjectured reference, not an established bound. Source ↗ |
Matrix multiplication · 8 milestones
| Date | Value | Evidence & attribution | Context & source |
|---|---|---|---|
| ≤ log₂ 7 | Published resultVolker Strassen | Strassen broke the cubic barrier with a recursive algorithm, giving an exponent of log₂(7), approximately 2.807355. Primary source ↗ | |
| ≈ 2.375477 | Published resultDon Coppersmith and Shmuel Winograd | The Coppersmith–Winograd method pushed the exponent below 2.376. This chart uses 2.375477 from Tao’s numerical reference ledger and the journal publication year, 1990; conference presentation was in 1987. Primary source ↗ · Supporting source ↗ | |
| < 2.3728639 | Published resultFrançois Le Gall | New analysis gave an exponent below 2.3728639. Posted and published at ISSAC in 2014. Primary source ↗ | |
| < 2.3728596 | Published resultJosh Alman and Virginia Vassilevska Williams | A stronger analysis lowered the exponent to 2.3728596. Preprint in 2020; SODA publication in 2021. Primary source ↗ | |
| < 2.371866 | Published resultRan Duan, Hongxun Wu, Renfei Zhou | The asymmetric hashing approach improved the exponent to below 2.371866. Preprint in 2022; FOCS publication in 2023. Primary source ↗ | |
| ≤ 2.371552 | Published resultVirginia Vassilevska Williams, Yinzhan Xu, Zixuan Xu, Renfei Zhou | The authors report an upper bound of 2.371552. Preprint in 2023; SODA publication in 2024. Primary source ↗ | |
| < 2.371339 | Published resultJosh Alman, Ran Duan, Virginia Vassilevska Williams, Yinzhan Xu, Zixuan Xu, Renfei Zhou | Further refinements yield an exponent below 2.371339. First posted in April 2024 and published at SODA in 2025. Primary source ↗ | |
| < 2.371177 | PreprintEmilien Dupont and collaborators | The preprint combines gradient optimization with an algorithm refined by AlphaEvolve. It reports an exponent below 2.371177 and rigorous verification of the resulting bound. Primary source ↗AI contribution: AlphaEvolve refined the optimization algorithm after gradient-based optimization had already improved the existing bound. This is distinct from small-matrix records by AlphaTensor. |
Ramsey numbers · 6 milestones
| Date | Value | Evidence & attribution | Context & source |
|---|---|---|---|
| 43–53 | Published resultBrendan McKay and Stanisław Radziszowski; lower bound by Geoffrey Exoo | The range is 43 through 53, inclusive. The authors’ later presentation records the 1992 upper bound and the subsequent improvements. Primary source ↗ | |
| 43–52 | Published resultBrendan McKay and Stanisław Radziszowski | The upper bound falls to 52. Historical date and value are recorded in McKay and Angeltveit’s 2017 presentation. Primary source ↗ | |
| 43–50 | Published resultBrendan McKay and Stanisław Radziszowski | The upper bound falls to 50, leaving eight possible integer answers. Historical date and value follow the authors’ presentation. Primary source ↗ | |
| 43–49 | Published resultBrendan McKay and Stanisław Radziszowski | The upper bound reaches 49. The next improvement in this selected history does not arrive until 2017. Primary source ↗ | |
| 43–48 | Published resultVigleik Angeltveit and Brendan McKay | The authors prove R(5,5) ≤ 48 with computer-assisted methods. The preprint appeared in 2017 and was published in 2018. Primary source ↗ | |
| 43–46 | Published resultVigleik Angeltveit and Brendan McKay | Mathematics, linear programming and extensive computation establish R(5,5) ≤ 46. The authors report independent replication. Preprint in 2024; journal publication in 2026. Primary source ↗ · Supporting source ↗ |
Odd perfect numbers · 6 milestones
| Date | Value | Evidence & attribution | Context & source |
|---|---|---|---|
| > 10^20 | Published resultHans-Joachim Kanold | Kanold’s 1957 lower bound of 10^20 is recorded in the introduction to Hagis’s original 1973 paper. This milestone is sourced through that later mathematical paper. Primary source ↗ | |
| > 10^50 | Published resultPeter Hagis Jr. | Hagis proves that an odd perfect number must exceed 10^50. The date is the publication year; this selected timeline omits some intermediate bounds. Primary source ↗ | |
| > 10^160 | Published resultRichard Brent and Graeme Cohen | The published paper raises the lower bound to 10^160 using restrictions on the prime factors of an odd perfect number. Primary source ↗ | |
| > 10^300 | Published resultRichard Brent, Graeme Cohen, Herman te Riele | The journal paper proves the 10^300 lower bound. A full technical report had already appeared in October 1989; this chart consistently uses the publication year. Primary source ↗ | |
| > 10^1500 | Published resultPascal Ochem and Michaël Rao | A computer-assisted proof establishes that any odd perfect number exceeds 10^1500. Published in Mathematics of Computation in 2012. Primary source ↗ | |
| > 10^2000 | Author reportPascal Ochem and Michaël Rao | The authors’ dated Montpellier presentation reports 10^2000 and explicitly labels the extension unpublished. It is displayed as an author report, separately from the published bound. Primary source ↗ |
Hadamard matrices · 3 milestones
| Date | Value | Evidence & attribution | Context & source |
|---|---|---|---|
| 13 gaps | Published constructionDragomir Ž. Đoković | A construction of order 764 leaves the 13 orders listed in the paper’s introduction unresolved through 2,000. The original list gives each order divided by four. Primary source ↗ | |
| 12 gaps | Published constructionDragomir Ž. Đoković, Oleg Golubitsky, Ilias S. Kotsireas | The paper constructs order 1,004 and lists the remaining 12 gaps. Preprint in January 2013; journal publication in 2014. Primary source ↗ | |
| 0 gaps | Author reportLevent Alpöge and collaborators; Claude contribution reported by Epoch | Epoch’s problem page records constructions for all previously unknown admissible orders through 2,000. Ion Nechita independently checked the matrix of order 668; this is not a proof of the general conjecture. Primary source ↗ · Supporting source ↗AI contribution: Epoch credits three human researchers working with an internal Claude model and explicitly marks the AI attribution provisional. |