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Announcing the first SHA-1 collision
Posted by Marc Stevens (CWI Amsterdam), Elie Bursztein (Google), Pierre Karpman (CWI Amsterdam), Ange Albertini (Google), Yarik Markov (Goog...
Topics /
Math & Algorithms
The most popular blog posts about math & algorithms on Hacker News. 2,831 posts.
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Bicycle
Interactive article explaining the physics of a bicycle.
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Airfoil
Interactive article explaining the physics of an airfoil and what makes airplanes fly
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Learning to Reason with LLMs
We are introducing OpenAI o1, a new large language model trained with reinforcement learning to perform complex reasoning. o1 thinks before it answers—it can produce a long internal chain of thought before responding to the user.
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4B If Statements
I recently stumbled upon this screenshot while researching social media on the train. Of course, it was followed by a cascade of spiteful comments, criticizing this fresh programmer’s attempt to solve a classical problem in computer science. The modulus operation.
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An OpenAI model has disproved a central conjecture in discrete geometry
An OpenAI model solved the 80-year-old unit distance problem, disproving a major conjecture in discrete geometry and marking a milestone in AI-driven mathematics.
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AI solves International Math Olympiad problems at silver medal level
Breakthrough models AlphaProof and AlphaGeometry 2 solve advanced reasoning problems in mathematics
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Sharing AI progress in mathematics
OpenAI publishes new results on open problems in mathematics from an internal frontier model and shares Lean proof formalizations and research details on GitHub.
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On the Navier–Stokes Millennium Prize Problem
We’re sharing an AI-generated solution to the Navier–Stokes Millennium Prize Problem, including a writeup and a formal proof in Lean.
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Mathematics for the Adventurous Self-Learner
For over six years now, I've been studying mathematics on my own in my spare time - working my way through books, exercises, and online courses. In this post I'll share what books and resources I've worked through and recommend and also tips for anyone who wants to go on a similar adventure.
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Fable 5.1 Solves the Cyphral Distich, a 370-year-old cipher
Explore Claude Fable 5.1's proposed solution to Sir Thomas Urquhart's Cyphral Distich, a centuries-old cipher, with the reasoning behind the result.
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How is NSA breaking so much crypto?
There have been rumors for years that the NSA can decrypt a significant fraction of encrypted Internet traffic. In 2012, James Bamford published an article quoting anonymous former NSA officials stating that the agency had achieved a “computing breakthrough” that gave them “the ability to crack current public…
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What Gödel Discovered
In 1931, a 25-year-old Kurt Gödel wrote a proof that turned mathematics upside down. The implication was so astounding, and his proof so elegant that it was...kind of funny. I wanted to share his discovery with you. Fair warning though, I’m not a mathematician; I’m a programmer. This means my understanding is…
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Visualizing Attention, a Transformer's Heart [video]
Demystifying attention, the key mechanism inside transformers and LLMs.
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Going Critical
Learn how things spread with playable simulations
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Knuth's Art of Computer Programming, V 4B, has gone into print
(I'm still looking for the catchiest way to appreciate the number MMXXVI. Exercise 7.2.1.6–122 gives 1234+.5*6+789 and -1+2345*6/7+8+9 and 1+(2/3)*45*67.8-9 ; other suggestions are welcome. The most elegant that I've seen so far is a beauty that was suggested independently by Arnaud Lefebvre and by Evangelos…
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A Programmer's Introduction to Mathematics
For the last four years I’ve been working on a book for programmers who want to learn mathematics. It’s finally done, and you can buy it today. The website for the book is pimbook.org, which has purchase links—paperback and ebook—and a preview of the first pages. You can see more snippets later in the book on the…
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Conterintuitive facts in mathematics, CS, and physics
It is possible to compute over encrypted data without access to the secret key: https://en.wikipedia.org/wiki/Homomorphic_encryption
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Too much efficiency makes everything worse (2022)
This blog is intended to be a place to share ideas and results that are too weird, incomplete, or off-topic to turn into an academic paper, but that I think may be important. Let me know what you think! Contact links to the left.
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I found two identical packs of Skittles among 468 packs
This article was discussed on Hacker News, and was cited in The Mathematical Gazette. Introduction This is a follow-up to a post from earlier this year discussing the likelihood of encountering two…
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Gravity is not a force – free-fall parabolas are straight lines in spacetime
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Ultimate Tic Tac Toe
Once at a picnic, I saw mathematicians crowding around the last game I would have expected: Tic-tac-toe.
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Deep image prior 'learns' on just one image
Deep convolutional networks have become a popular tool for image generation and restoration. Generally, their excellent performance is imputed to their ability to learn realistic image priors from a large number of example images. In this paper, we show that, on the contrary, the structure of a generator network is…
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Markov Chains Explained Visually (2014)
Markov chains, named after Andrey Markov , are mathematical systems that hop from one "state" (a situation or set of values) to another. For example, if you made a Markov chain model of a baby's behavior, you might include "playing," "eating", "sleeping," and "crying" as states, which together with other behaviors…
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Formalizing Fermat's Last Theorem
Anthropic is an AI safety and research company that's working to build reliable, interpretable, and steerable AI systems.
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