Kakeya conjecture — #38

Fourier restriction conjecture — #46

Local smoothing for wave equations — #187

Bochner–Riesz conjecture — #190

Exponential Time Hypothesis — #43

Strong Exponential Time Hypothesis — #63

Gemini 4 Argon: https://x.com/LechMazur/status/2111255403216384046 GPT-6 Pro: https://x.com/LechMazur/status/2111255536666644932 Opus 5.5: https://x.com/LechMazur/status/2111258566539493402

  +----------------------------------------------------+------+---------+-----------------+
  | Category                                           | Full | Partial | Matched / total |
  +----------------------------------------------------+------+---------+-----------------+
  | Geometry and topology                              |   25 |       7 |         32 / 74 |
  | Algebra, representation and category theory        |   17 |       2 |         19 / 53 |
  | Analysis and PDE                                   |   11 |       6 |         17 / 40 |
  | Number theory and arithmetic geometry              |    4 |      13 |        17 / 117 |
  | Probability, ergodic theory and dynamics           |   11 |       5 |         16 / 37 |
  | Combinatorics and discrete geometry                |    7 |       2 |          9 / 34 |
  | Theoretical computer science                       |    4 |       4 |          8 / 57 |
  | Mathematical physics                               |    5 |       1 |          6 / 19 |
  | Applied and computational mathematics              |    2 |       2 |           4 / 8 |
  | Quantum information and computation                |    2 |       1 |          3 / 17 |
  | Cryptography, coding, information and optimization |    1 |       1 |          2 / 26 |
  | Logic, foundations and set theory                  |    1 |       1 |          2 / 18 |
  +----------------------------------------------------+------+---------+-----------------+
  | Total                                              |   90 |      45 | 135 / 500 (27%) |
  +----------------------------------------------------+------+---------+-----------------+

The highest ranked would be:

| 22 | Hilbert’s tenth problem over ℚ |

| 29 | Unique Games |

| 31 | Anderson-model extended states |

| 37 | Spacetime Penrose inequality |

| 48 | Nonexistence of Landau–Siegel zeros |

| 52 | Baum–Connes |

| 78 | Abundance |

| 80 | Hadwiger |

| 87 | Bose–Einstein condensation |

| 92 | Two-dimensional entanglement area law |

Right now, I'm having LLMs audit the actual math in claimed arXiv solutions because despite its policy changes, arXiv is still a dumping ground. The audits have already found six faulty proofs that caused status issues for problems that should still clearly be fully open.

But what's crazy is that within the last day or so, we've also gotten LLM-assisted solutions to #95, the Kannan–Lovász–Simonovits (KLS) conjecture, by three different authors in parallel (all extending Song–Zhang's key criterion introduced on Oct. 1), #278, the Mumford–Shah conjecture, and #227, Zauner's conjecture on SIC-POVM existence in every dimension, which also represents a major claimed advance on Hilbert's twelfth problem (#36) for real quadratic fields.

This is likely because OpenAI's solutions to 100 open conjectures are expected to drop any day, so everyone is in a hurry not to get scooped.

This is all you need to read and understand for Anthropic's FLT formalization:

  import Mathlib
  import Theorems.Thm_fermat_last_theorem

  /-- Solution side: the same statement, binder for binder, proved by this tree's `fermat_last_theorem`. -/
  theorem FLT_for_comparator (n : ℕ) (hn : 3 ≤ n) (a b c : ℕ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) :
    a ^ n + b ^ n ≠ c ^ n :=
  fermat_last_theorem n hn a b c ha hb hc

  /-- Mathlib's named proposition, by the one-line bridge from the elementary statement
  (the bridge is restated inline so that this file depends only on `Theorems.Thm_fermat_last_theorem`). -/
  theorem FLT_mathlib_for_comparator : FermatLastTheorem :=
  fun n hn a b c ha hb hc => fermat_last_theorem n hn a b c (Nat.pos_of_ne_zero ha) (Nat.pos_of_ne_zero hb) (Nat.pos_of_ne_zero hc)

"Problems solved before a model's training cutoff can be filtered out, and all models compared on the remaining problems" means that the problems an older model actually solved are the ones that get filtered out, while the remaining problems are the ones it already tried and failed on. So older models end up with 0s on the filtered set and you can't really use this to compare new models to older ones.

Also, since these are known public problems, you can't stop people from spending far more than your arbitrary time and $ limits on them. So the number of clean problems will go down over time.

Comments URL: https://news.ycombinator.com/item?id=48996871

Points: 2

# Comments: 0