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Plan used by Anthropic’s Claude to formalize Fermat’s Last Theorem; source: Anthropic Introduction In 1637, French mathematician Pierre de Fermat, while perusing a copy of Arithmetica, an ancient mathematical text by the Greek mathematician Diophantus, wrote in the margin (English translation): “It is impossible to separate a cube into two cubes, or a fourth power into two fourth powers, or in general, any power higher than the second, into two like powers. I have discovered a truly marvelous proof of this, which this margin is too narrow to contain.” In modern mathematical parlance, “No positive integers a, b, c Continue reading AI software completes a formal proof of Fermat’s Last Theorem

The march of human progress There is much to celebrate in modern society: Crime rates are declining worldwide; worldwide extreme poverty continues to fall; worldwide life expectancy continues to rise; computer and information technologies continue their remarkable forward march; artificial intelligence is on the rise, which, while carrying some risks, also holds great promise; and advances in science and medicine continue on many fronts, far too numerous to list. See this previous Math Scholar article for more details. A crisis in education However, there is one arena of concern in today’s society that certainly deserves more attention: the education Continue reading The great dumbing down of society

MathJax = { tex: {inlineMath: [['$', '$'], ['\\(', '\\)']] }, svg: {fontCache: 'global'}}; Credit: Vatican Museum 2023-2025: Can chatbots prove mathematical theorems? Three years ago (23 Feb 2023), the present author tested ChatGPT, which then was the state-of-the-art in AI-based large language models (LLMs), with requests to produce rigorous mathematical proofs for four well-known but nontrivial theorems: A general angle cannot be trisected with ruler and compass. $\pi$ is irrational. $\pi$ is transcendental. Every algebraic equation with integer coefficients has a root in the complex numbers. The results, frankly, were abysmal. For example, ChatGPT attempted to prove the Continue reading How effective are current AI models on mathematical research problems?

credit: Vera C. Rubin observatory Where are the extraterrestrial civilizations? (Fermi’s paradox) Updated 23 May 2026 (c) 2026 Introduction In 1950, while having lunch with colleagues Edward Teller and Herbert York, who were chatting about a recent cartoon in the New Yorker depicting aliens, physicist Enrico Fermi suddenly blurted out, “Where is everybody?,” a question now known as Fermi’s paradox. This article presents background on Fermi’s paradox, explains why many of the proposed solutions are NOT viable, and mentions a few promising new results and directions. Behind Fermi’s question was this line of reasoning: (a) Given the vast Continue reading Where are the extraterrestrials? Fermi’s paradox, diversity and the origin of life

Yes, in just a few days we will once again celebrate Pi Day (3/14 in North American date notation). Our custom on this site has been to offer a crossword puzzle on a math or Pi theme. This year we offer two puzzles (take your pick or do both). Some notes: 1. Each puzzle is written to standards of The New York Times daily puzzles. 2. In each puzzle, a handful of the squares contain two letters. 3. No mathematical knowledge is required to solve either puzzle. 4. The overall difficulty of these puzzles is Tuesday or Wednesday on the Continue reading PiDay 2026: Double crossword