How effective are current AI models on mathematical research problems?

MathJax = { tex: {inlineMath: [[‘$’, ‘$’], [‘\\(‘, ‘\\)’]] }, svg: {fontCache: ‘global’}};

Credit: Vatican Museum

2023-2025: Can chatbots prove mathematical theorems?

Two 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?

Where are the extraterrestrials? Fermi’s paradox, diversity and the origin of life

credit: Vera C. Rubin observatory

Where are the extraterrestrial civilizations? (Fermi’s paradox) Updated 4 April 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