Category Archives: Mathematics

AI disclosures done right

Posted on September 27, 2026 by Persiflage

Another shiny new AI proof, this time a proof by Constantin Kogler of the Odlyzko–Poonen conjecture which says that a polynomial with coefficients chosen randomly from \(\{0,1\}\) is irreducible. What I want to emphasize on this occasion is the exemplary … Continue reading →

Posted in Mathematics | Tagged , , , | 2 Comments

Roots of Unity

Posted on September 26, 2026 by Persiflage

This post is a follow-up on a previous post on the abelian house. For an algebraic integer \(\alpha\), the house \(\overline{|\alpha|}\) is the absolute value of the largest conjugate of \(\alpha.\) Raphael Robinson made a number of conjectures concerning cyclotomic … Continue reading →

Posted in Mathematics | Tagged , , , , , , , , , , , , | Leave a comment

zeta(5) is irrational

Posted on September 24, 2026 by Persiflage

So one day after an amazing result by humans, the computers have struck back! This time, a proof that \(\zeta(5)\) is irrational. The paper I saw gives me the impression of being almost entirely AI generated (if so, it then … Continue reading →

Posted in Mathematics | Tagged , , , , , | 2 Comments

Real quadratic fields and finite quantum dilogarithms I

Posted on September 22, 2026 by Persiflage

Danylo Radchenko and Campbell Wheeler have posted an extraordinary new paper in which they prove that Stark units for real quadratic fields are algebraic numbers. (Not yet a Shimura reciprocity law that shows these generate abelian extensions but hey, this … Continue reading →

Posted in Mathematics | Tagged , , , , , , , | 5 Comments

An even PSL_2(F_23) Galois extension

Posted on September 20, 2026 by Persiflage

I was wrong! I have said on a number of occasions that my favourite family of groups for the inverse Galois problem is \(G=\mathrm{SL}_2(\mathbf{F}_p)\). This is still true, and the reason is still the same; the fact that \(G\) has … Continue reading →

Posted in Mathematics, Uncategorized | Tagged , , , , , , , | Leave a comment

Ways to improve Mathematics (an introduction)

Posted on September 17, 2026 by Persiflage

The goal of this continuing sequence of posts is to recognize the reality in how our subject is changing and how we should adapt. Even in a hypothetical world where AI was infallible, essentially omniscient, benevolent, a wonderful expositor, and … Continue reading →

Posted in Mathematics | Tagged , , , , , , , | 8 Comments

Look, Mom, I pressed a button! (Go edition)

Posted on September 9, 2026 by Persiflage

Apropos of nothing, I was reminded recently of a fascinating story I heard from Geordie Williamson about AI and go — perhaps this story was the original source. It is a story about how a technology which has the power … Continue reading →

Posted in Mathematics | Tagged , , | 5 Comments

Look, Mom, I pressed a button!

Posted on September 5, 2026 by Persiflage

I have recently heard a few extraordinary opinions about how mathematicians should use AI. One I would paraphrase as “professional mathematicians should never use AI for any problem not in their narrowly defined (by whom?) research program”, which seems ridiculous … Continue reading →

Posted in AI, Mathematics | Tagged , , , , , , , | 11 Comments

OpenAI, updated

Posted on August 23, 2026 by Persiflage

An update on this post, from my inside sources: word on the math streets of SF is that the OpenAI team tried something like 500 problems to get their 10 solutions. I don’t know how much of an insider this … Continue reading →

Posted in Mathematics | Tagged , , | 6 Comments

The inverse Galois challenge, part II

Posted on August 16, 2026 by Persiflage

This is a sequence to this post. The SAIR competition (Round I) has been completed! 98.4% of the possible signatures were obtained, with only 39 non-solvable cases missing. Some thoughts. First, my timing in the last post of dissing the … Continue reading →

Posted in Mathematics | Tagged , , , , , , , , , , , , | 3 Comments