Claude AI took 650 attempts to solve a Riemann-hypothesis-related problem, then beat the human record. The breakthrough came from simple encouragement prompts like "keep going, believe in yourself," and the AI worked for 37 minutes in silence before finding the result. A formally verified proof backs the claim.
What is happening? An unreleased version of Claude was asked to solve a long-standing mathematical problem, the Riemann hypothesis. Simplified, it is a statement about the distribution of prime numbers. Devilishly difficult. No human has ever been able to prove it. Did it solve it? Nope. So what did it do then? Well, it actually was able to improve a related bound beyond human record. It pushed humanity further. Sort of. I think that is incredible. Now, there is the mathematical side. Many mathematicians I heard seem to be both surprised and impressed by the results. Some call it a massive leap forward. I ran the verification myself, but I am just a student looking to learn, and I am not qualified to speak more about the mathematical side of it. But the AI side, hoo-hoo, is incredible. 3 things that happened that I found Super interesting. One surely exquisite mathematical prompting was done, right? No, not really. Here's what happened. First, a non-mathematician person prompted this AI. Wow. Okay. The first 650 tries did not work. Now hold on to your papers, Fellow Scholars, quoting, throughout this process, Jared's input was mostly limited to sending Claude messages of encouragement. Mostly variants of keep Keep going and believe in yourself. What? He had to give words of encouragement to an AI to keep going, and it succeeded! Perhaps in the future, the most powerful mathematical proofs will not be written by geniuses. They will be written by life coaches. What a time to be alive! And get this, it's not the first time this has happened. Quoting, a prompt including similar encouragement was used to help Claude disprove the Jacobian conjecture. Okay. But I was even more surprised about 2 more things. Dear Fellow Scholars, this is Two Minute Papers with Dr. Károly Zsolnai-Fehér. 2, the full technical paper is available, but it's pretty tough. Of course. So much so that they asked the AI to explain its findings. So it did. And in the meantime, a formalized version of the proof is also available, which can be automatically verified. You can even run it yourself right now. Now, something I don't think you hear too much about elsewhere. I went through more than 100 pages of transcripts, and some super cool tidbits from the journey. Claude had internet access in general, but did not need to use it during the key breakthrough run. Ooh! Claude went down many wrong roads first, but was able to learn from them and recover. Then, the first crucial result appeared after about 37 minutes of radio silence. Imagine how tense that must have been. And then, pop! And the AI itself also thought the result is suspiciously good. It said quoting, too strong to be new. Absolutely crazy. And 3, even Claude was surprised about its finding. And it was skeptical at first. They also say perhaps Claude itself also underestimates the rate of AI progress. But, an AI that is surprised? Of course, this does not mean a human-like surprise. It simply reflects patterns learned from us during training. But still, I feel like we are living in a new world where sentences that didn't used to make sense now suddenly do. And in a world where these AI systems built by human ingenuity are now pushing humanity forward. What a time to be alive! We need new tools for the era of LLMs, and Weights Biases now has Weave, a lightweight toolkit to confidently iterate on LLM applications. Use traces to debug how data flows through each step of your app, and use evaluations to measure your progress. It is the best. Try it out now at wnb.me/papers or click the link in the description below.
