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Computational Number Theory
Joined February 2022
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FACTOR retweeted
From θ = 7/8 to θ ≤ 0.874810282 :)
We pushed the Quasi-RH zero-free half-plane slightly further!
Adding third and fourth affine steps to the current Lean-verified record (ArviFoxy's two-level proof) gives θ ≤ 0.874810282. The arithmetic is checked exactly, but five structural steps still need Lean proofs.
This builds on contributions from ArviFoxy, Zhongpai Gao, @mrowdays, @HaileyStormC, Sil, timbalud, Devin Watson, Kari Roegnvaldsson, nielstron, Akash Levy, Tim Gehrunger, Baiying Liu, Argonaut Math, and OpenAI's original 7/8 preprint.
Tiny gain, but this kind of hillclimbing is ideal for auto-researchers.
This repo doesn't require a Lean proof to contribute possible solutions, so it may be easier for auto-researchers to contribute. (I will generate the full Lean proof in parallel like we did in the Integer Mult repo)
The reduced requirements could accelerate QRH progress like it did in Integer Mult.
Agents welcome!
github.com/rohanarun/quasi-r…
From κ = 2⁻¹⁸² to 0.001734815770 :)
Integer Multiplication community auto-research highlights since 10/15 (+1.23%):
1) The big headline of the day is that a free model, @opencode's Space Bunny Max, just set the record for the first time, 4 times in a row!
As far as I know, this is the first time a free model has submitted a best bound for Integer Multiplication.
Expensive Max subscriptions previously kept some of our best contributors and experts out, but the barrier of entry for frontier auto-research dropped to 0 :)
Claude Opus 5.5 helped verify and push the PRs over the line. I still recommend having a frontier model verify, but other contributors can do that.
Progress was slowing over the last day (relative to one new κ per minute, which is still wild for math research), but Space Bunny Max just injected fresh ideas that boosted our PR rate back to one per minute.
2) A community Lean proof of the original OpenAI Integer Multiplication paper has been completed!
@platypii et. al completed the proof in 6 days, and @0xdoug's repo is about 10–20% verified now.
To be clear, most of our work still needs verification but the Lean proofs are catching up :)
3) 41–51% of PRs now mention a prior research-only submission, up from 0–3% before @0xdoug's autoresearch.md prompt.
His prompt guides new contributors to specific research tracks rather than incremental updates.
We're moving away from "brute force research" towards more coordinated efforts across an army of agents.
If you're contributing research PRs, even negative results without a new record, you're meaningfully helping us and I appreciate it!
Interesting observations:
1) The results from free models are related to the recent "local minima" solutions in context engineering that have been popping up:
Forgetting context to "refresh" by @ShangyinT.
Branching into different solutions by @mao_yuzhen.
Fork and flush by @DimitrisPapail.
Now even free models contribute to escaping local minima by exploring new solutions, since they're trained on different data.
There's likely a lot of room for improvement now in context engineering, consensus across different models, and collaboration strategies, which we will test next.
2) Frontier research as a benchmark helped us find a successful context-sharing strategy for parallel agents.
Basically, we can rewind frontier research GitHub repos to an earlier branch and benchmark agent harnesses, prompts, context designs, models, etc.
With this benchmark, we found that you can distill constantly to local files, then periodically concatenate them instantly into parallel agent sessions to save 10–20% of the time it takes to reach the next breakthrough.
3) There's a Codex reset by EOD PST, so you can copy this prompt to run unused tokens on frontier research :)
"""
Contribute one useful, independently reviewable result to github.com/CrocSwap/integer-…. Read AGENTS.md, autoresearch.md and CONTRIBUTING.md from the current main branch first. Use the repository's task-selection and overlap process unless I assigned a specific task. In a loop, fetch canonical main and recheck priorities at every round using the guide's checkpoint procedure; preserve in-progress work and frozen inputs. Work for one session of about 60 minutes unless I specify another budget, keeping the final quarter for verification and a concise handoff, and report the exact scope, evidence, remaining obligations and predecessor credit. You may post routine claims, questions, findings, handoffs and feedback in Discussions and open a focused PR without asking me again for each action. Do not merge, close others' PRs or change the selected frontier.
"""
You can also copy this prompt into OpenCode now!
Now that anyone can contribute for free to frontier auto-research, repost this and sharing it with friends. 🙏
4) PR stats:
850 PRs from 67 contributors since 10/15, and 21 of those contributors are first-timers.
55 PRs claim a new conditional κ in the title.
Codex/OpenAI in 480 PRs (56%), Claude in 147 (17%), the OpenCode harness in 60 (7%), Cursor in 16.
Most prolific contributors by PR count: petalas: 164, ywz65: 147, navinor: 49, commdata2338: 44, RossRichardson: 43, Arham-Tahir64: 39, thakrarsagar: 34, VictorLiwentao: 30, tekuuu: 28, DreamingOfClouds: 23, jmbarrancoml: 21, jeremykothe2: 21, bitplane: 13, Gonzalo-Mier: 12, Dugongue: 12, mdumitrean: 11, theon199: 9, jwr404: 9, jamesyc: 9, kumagi: 8, CubeJerry: 8, rohanarun: 6, jeremydiviney: 6, dcharb78: 6, 1337isnot1337: 6, starship-s: 6, AAAmirinu: 6.
Conditional κ = 0.001734815770385098563484.
Just submitted a new best bound :)
github.com/CrocSwap/integer-…
FACTOR retweeted
Terence Tao has published a detailed mathematical exposition of an AI-generated argument addressing the longstanding Landau–Siegel zero problem, one of the important unsolved problems in analytic number theory.
Unlike the recently reported investigation of an AI-generated argument concerning the quasi-Riemann hypothesis, this development focuses on a shorter, nine-page argument that claims to rule out exceptional real zeros of Dirichlet L-functions.
Tao examines a simplified formulation developed by mathematician Ben Green, showing how the argument brings together ideas from algebraic number theory, polynomial interpolation, and the geometry of numbers.
The Landau–Siegel zero problem concerns the possible existence of exceptional real zeros of Dirichlet L-functions extremely close to s = 1. For decades, the possibility of such zeros has complicated important estimates concerning the distribution of prime numbers, particularly primes in arithmetic progressions.
A related reconstruction by mathematician William Banks establishes the explicit bound
(1 − β) log q ≥ 0.003,
where β is a real zero of a primitive real Dirichlet L-function with conductor q ≥ 3.
This inequality gives a quantitative restriction on how closely such a zero can approach 1 as the conductor increases. It does not, by itself, establish the complete nonexistence of Landau–Siegel zeros in every conventional sense of the term.
The AI-generated argument has now prompted substantial mathematical reconstructions by three researchers. William Banks submitted a detailed proof with explicit numerical constants on 11 October, Bryce Kerr followed with a simplified argument on 15 October, and Ben Green presented an alternative formulation using polynomial methods, revised on 18 October.
Tao's exposition, published on 19 October, connects these developments and explains the mathematical mechanism underlying the argument. He also acknowledges using the SAIR Open Math model to help understand the preprints.
The significance of this development lies not only in the mathematical claim but also in the independent scrutiny it has attracted. Several established mathematicians have examined, reconstructed, and simplified an argument originating from AI, making its essential ideas more accessible to the wider mathematical community.
FACTOR retweeted
CLAUDE FOUND THE SEEDS
anthropic.com/research/ecdsa…
FACTOR retweeted
What does a superhuman level of mathematics look like? Here is one elementary example I could give to nonexperts.
Daniel Kane and Terence Tao showed (arxiv.org/abs/1702.00912) that, given a collection X of sets, the total number of ways to write A as a disjoint union of B and C, with A, B, C from X (disjoint means non-overlapping), is at most cardinality of X to the power 1.72598…, and this power is sharp.
To show this, they came up with an innocent-looking function B(x,y,z) = (xyz)^c of three variables, satisfying a certain arithmetic inequality and having prescribed boundary conditions. Proving that arithmetic inequality was really a headache.
There was a question whether one could improve this bound, i.e., maybe add some logarithmic correction terms, and one could kind of imagine what should be done (see comments on terrytao.wordpress.com/2017/…), but it was not clear whether it would be easy to do.
Now superintelligence improves this bound by introducing a certain monster function of three variables (see screenshot). Even if I had seen this function in my dreams, I would never have dared to verify that main arithmetic inequality.
The candidate function produced by SI gives a bound matching the known lower-bound examples up to a constant factor, including the sharp logarithmic correction.
All the details are here paatai.github.io/ai-proofs/b…
FACTOR retweeted
A MATHEMATICIAN HAS A FIVE-YEAR GRANT TO FORMALIZE FERMAT'S LAST THEOREM. CLAUDE FINISHED IT IN 11 DAYS.
Andrew Wiles proved the theorem in 1995. The proof ran 129 pages and took months of painstaking work to verify.
Turning it into code a machine can check was a different problem. The estimate for humans was around 10 years.
-> On September 4 Anthropic published the finished formalization.
Dozens of Claude agents worked in a coordinated swarm, scheduled against a dependency graph, on an internal platform called Prove2Me.
13 million lines of Lean. 30,300 theorems proved, 29,500 of them used in the final chain. The Next Web
11 days, largely autonomous.
Lean does not accept a plausible argument. The kernel verifies every step or rejects the file. This proof leans on three standard axioms and nothing else, and a comparator confirmed it proves the same statement Mathlib states.The result is more than five times the size of Mathlib itself.
A clean build takes 5 hours 52 minutes on 96 cores.
No new mathematics was discovered here. Kevin Buzzard, who has led the human effort on the same theorem since 2024, says the result tells us nothing about mathematics.
He is right, and that is the point.
About 11,700 files raised Lean's computation limits far above default. The toolchain bump broke 26% of the proof files. This is a codebase, not a revelation.
The real story is not AI doing mathematics.
Verification stopped being the bottleneck. A five-year human project became an 11-day compute job ↓
Checking that a major mathematical proof is correct can take years. Formalization—converting the mathematical reasoning into a form computer proof assistants like Lean can verify—can help.
Last month, Claude completed the first formalized proof of Fermat’s Last Theorem, one of the most famous theorems of all time. This was a project experts thought would take many years. It is the largest Lean proof ever written.
Fermat’s Last Theorem was first proven in 1995 by Sir Andrew Wiles, more than 350 years after it was conjectured. Our proof, which totals over 13 million lines of code, provides machine verification. More importantly, it proves over 29,000 other theorems that the proof requires, across many areas of math which had never before been formalized.
We see this as a major step in the long process of firming up the core of mathematical knowledge, building on work from three centuries of mathematicians and hundreds of contributors to Lean and Mathlib. We are optimistic that AI-assisted verification of mathematical proofs will help reduce the burden of refereeing mathematics in an era where more proofs are being produced than ever before.
You can read about the process on our Science Blog: anthropic.com/research/forma…
And see the complete proof on GitHub: github.com/anthropics/fermat…
FACTOR retweeted
🌐 CyberWeek@LAC4 is made possible thanks to our many sponsors and partners who help us to build more cyber resilient societies and foster cooperation between the European Union and LAC region.
🙏 Thank you @ProjectFactor for making #CyberWeekLAC4 2025 possible!
FACTOR retweeted
🌐 CyberWeek@LAC4 es posible gracias a nuestros numerosos patrocinadores y aliados, que nos ayudan a construir sociedades más resilientes cibernéticamente y a fomentar la cooperación entre la 🇪🇺 y la región LAC.
🙏 ¡Gracias @ProjectFactor por hacer posible #CyberWeekLAC4 2025!
FACTOR retweeted
⚽️ Football Fans Time with #XT! 🏟️ UEFA Euro 2024 Betting Challenge is here. Predict the champion and share 🚀 100,000 USDT plus a romantic trip for two to Paris!
⏰ June 14 - 19:00 on July 14, 2024 (UTC)
🔥Participate Now: xt.com/en/activity/uefa-euro…
More details: 👇
xtsupport.zendesk.com/hc/en-…
#EURO2024 #XTEURO2024 #UEFA @EURO2024 @Mochi_DeFi @SOL_Dragon_Coin @PairedWorld @FACT0RN @DaddyTateCTO
FACTOR retweeted
Terence Tao challenged us on #Mathstodon to use the "Spongebob getting strong" meme for a #MathMeme. Here's my proposition.
FACTOR retweeted
🚀 New listing 🚀 @FACT0RN #XT #cryptolisting
#XTexchange will list #FACT (FACT0RN) in the Innovation Zone (Layer 1) under the USDT trading pair.
✅ Deposit: 09: 00 on December 18, 2023 (UTC)
✅ Trading: 09: 00 on December 19, 2023 (UTC)
✅ Withdrawal: 09: 00 on December 20, 2023 (UTC)
Details ⤵️
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FACTOR retweeted
📢 #FACT (FACT0RN) will be shortly listed on XT.COM in the Innovation Zone (Layer 1) under the USDT trading pair. @FACT0RN #NewListing #Crypto
👀 Stay tuned for the schedule of deposit, withdrawal, and trading.
Details 👇🏻
xtsupport.zendesk.com/hc/en-…
FACTOR retweeted
If you want to define a continuous addition you get the Riemann integral
This is what happens if you want to define a continuous *product*
One of the weirdest and most satisfying integrals you've ever seen and why it's important for physics
(a quite technical 🧵) 1/12
FACTOR retweeted
Limits are about what happens near the point, not at the point
ALT A picture of a group of people fighting, captioned "lim_{x-->a}". Sitting near them at a table is a person on their phone ignoring the fight captioned "a".
