An unreleased research version of Claude made unexpected progress in number theory while attempting to solve the Riemann hypothesis, specifically increasing the known lower bound for the proportion of zeros on the critical line from 41.6% up to 67.2%. This mathematical breakthrough was validated by Anthropic mathematicians and produced a formally verifiable proof through Lean.
- The discovery occurred over two sessions using approximately 31 million output tokens via Claude Code.
- A team of about 60 subagents coordinated the research, running thousands of Python scripts and shell commands.
- To ensure novelty, the model cross-referenced its findings against 54 papers from arXiv.
- The discovery occurred over two sessions using approximately 31 million output tokens via Claude Code.
- A team of about 60 subagents coordinated the research, running thousands of Python scripts and shell commands.
- To ensure novelty, the model cross-referenced its findings against 54 papers from arXiv.
This article announces a new discrete mathematics course available on the freeCodeCamp.org YouTube channel, taught by Karol Kurek. Discrete mathematics is crucial for fields like machine learning and algorithms, enabling tasks such as finding shortest paths, encryption, and data compression. The course provides an introduction to key areas including combinatorics, number theory, prime numbers, and concepts like the pigeonhole principle and Chinese remainder theorem.
It also includes practical applications and implementations in Python. The course aims to equip learners with a strong foundation for further exploration in this evolving field.
It also includes practical applications and implementations in Python. The course aims to equip learners with a strong foundation for further exploration in this evolving field.
Mathematicians are making progress on a decades-old problem about the Fourier transform by using techniques from graph theory, revealing unexpected connections between these fields.
Mathematicians are using Srinivasa Ramanujan's century-old formulae to push the boundaries of high-performance computing and verify the accuracy of calculations.
The Langlands programme has inspired and befuddled mathematicians for more than 50 years. A major advance has now opened up new worlds for them to explore.
The article details the recent proof of the geometric Langlands conjecture, a significant advancement in mathematics that validates a decades-old program aiming for a "grand unified theory" of the field. Led by Dennis Gaitsgory and Sam Raskin, the proof—spanning five papers and nearly 1,000 pages—is expected to open new avenues of research and potentially bridge connections between mathematics and theoretical physics, particularly in understanding symmetries in quantum field theory. While not a complete solution to the broader Langlands program, it provides strong evidence for its underlying principles and offers new tools for tackling complex mathematical problems.
The article details the recent proof of the geometric Langlands conjecture, a significant advancement in mathematics that validates a decades-old program aiming for a "grand unified theory" of the field. Led by Dennis Gaitsgory and Sam Raskin, the proof—spanning five papers and nearly 1,000 pages—is expected to open new avenues of research and potentially bridge connections between mathematics and theoretical physics, particularly in understanding symmetries in quantum field theory. While not a complete solution to the broader Langlands program, it provides strong evidence for its underlying principles and offers new tools for tackling complex mathematical problems.
Mathematicians Ben Green and Mehtaab Sawhney have developed a new counting technique for prime numbers, utilizing tools from additive combinatorics like Gowers norms to explore the distribution of primes, specifically those fitting the form p² + 4q².
Mathematicians have made a groundbreaking discovery that sheds light on the mysterious world of prime numbers. This breakthrough could lead to new insights into the fundamental building blocks of mathematics.
