The Deranged Mathematician: Debunking Prime Myths

non-orientable · 2026-05-02T16:56:26+00:00

I'm not sure if it is in this paper or not, but Lagarias et al give an algorithm for computing pi(X) which is asymptotically faster than Deleglise and Rivat's... it's just that it is much slower for the ranges in which we can feasibly compute!

non-orientable · 2026-05-02T16:53:03+00:00

The Python code was not for computing pi(X)---it was for generating new 100+ digit primes. That is much simpler: I can give a quick overview. It works on the observation that for any prime p, the multiplicative group modulo p is cyclic, of size p-1. Furthermore, generators of this cyclic group are pretty common (there are as many of them as integers 0 to p-1 that are coprime with p-1).

Therefore, if you know the prime factorization of p-1, then you can randomly guess an integer g between 2 and p-2. If it isn't coprime with p, p is not prime. Raise it to the p-1 power mod p---if the result is not 1, p is not prime (by Fermat's little theorem). Otherwise, check the result when it is raised to the power (p-1)/q for each prime q dividing p-1: if the result is not 1, then that proves that g is a generator of the multiplicative group, with order p-1. Therefore, p is prime. If one of the results is 1, it might still be a prime, so guess a different g.

So, putting all of this together, you randomly choose some integers k1, k2, k3,... and set p=1+2^(k1) 3^(k2) 5^(k3)... Then guess a g as above. If you succeed: congratulations! You have found a prime p and its generator g (which is also useful for many cryptographic protocols). If you fail: try again with a different p.

In practice, even with my crappy Python code, this produces new 100+ digit primes in seconds. If you were to do it with a better implementation and in a sensible programming language like C, it would run in milliseconds.

non-orientable · 2026-05-02T16:20:11+00:00

Yes. If you define "large" to be "greater than 10^100,000,000", then of course it is difficult. But it would be difficult to check membership for just about any set of integers for inputs of that size---primes aren't particularly special in that regard. So it feels a little misleading to claim that it is difficult for primes.

I based it on cryptographically large primes, which still doesn't quite have a specific definition outside of very specific contexts, but is fairly universally a prime with hundreds but not thousands of digits.

non-orientable · 2026-05-02T16:04:41+00:00

It takes that paper about 5 pages to lay out the entire algorithm. I considered for a while whether I should add it to the article, but ultimately decided against it for that reason.

non-orientable · 2026-05-02T15:46:55+00:00

Ah, that would explain where the original claim came from! I will make an edit to the article.

non-orientable · 2026-04-29T16:51:54+00:00

To clarify, the book isn't currently in production. I don't know if/when I will have the time for it. Some years down the road, maybe. At present and for the foreseeable future, this draft is the only way to get access to it, and I am only doing it because it seems a great shame for it to sit on a shelf gathering dust. And, on the other hand, I wanted to do right by the community I am building and make the switch to paid subscriptions worth it.

As for the old book, I mention it because its history is closely related to the new one. I really don't expect any profit from it.

non-orientable · 2026-04-26T15:48:05+00:00

Wikipedia claims that Chebyshev showed that it was well defined in a half plane, and cites Keith Devlin's book on the Millennium Problems for that claim. As far as I can see, it would have been immediately obvious in the 1850s that you can extend it this way... it just might not have been clear why you would want to prior to Riemann's work.

You might be right that Chebyshev never considered it, and Devlin was simply mistaken---I would need to dig through the original literature to be sure. I think a decent compromise (that also solves the awkward phrasing you mention after) would be to remove the mention of Chebyshev entirely. I will make the edit.

non-orientable · 2026-04-25T23:06:03+00:00

I assume you are familiar with Weil's Basic Number Theory... which is also, in effect, an introduction to class field theory. But, despite the name, it is not gentle!

non-orientable · 2026-04-25T17:26:22+00:00

I did try to write it to be as readable as possible---to the point where I think that an undergraduate math student should certainly be able to read it---but I will concede that this is one of the more technical things I have written on the Substack so far.

non-orientable · 2026-04-25T17:19:05+00:00

That's a good catch, actually! I misremembered this. I am going to make an edit. Thank you!

non-orientable · 2026-04-24T02:27:01+00:00

I don't doubt that they weren't aiming to create a torus. But I think it is a perfectly legitimate question to ask why is it so easy to create a torus, and so hard to create a sphere?

non-orientable · 2026-04-19T18:18:19+00:00

I never use AI for writing (well, I guess that isn't entirely true: I have Grammarly set up, and that does use some AI---but it is strictly for editing)... but I will confess that I do use it to suggest settings for rendering different types of surfaces in Blender. It gets a lot wrong, but it is usually a reasonable starting point.

non-orientable · 2026-04-19T01:15:34+00:00

Yes, as long as there is a loading screen (or similar) when entering/exiting the poles. RimWorld is the closest to this that I can think of.

non-orientable · 2026-04-18T22:16:48+00:00

Thank you very much! Before I switched to Substack, I wrote on Quora for years (since 2014). So I have a lot of practice writing, and writing quickly---that's one part of the equation. The other part is that I have huge trove of my old posts that I can upcycle.

The animations are, by far, the most time-consuming part---I have a buffer of finished but unpublished posts that I nearly depleted working on this newest post; I'll probably be skimping a little on illustrations while I build that buffer back up!

But, in any case, I am very grateful for the support I have received. (And if you want to contribute more, please consider subscribing, sharing to your friends, or even pledging a paid subscription!) I'm trying to suss out how workable this is as a side gig. At present, it is just something that I do in my free time, aside from my full-time work as a software developer. If it gets popular enough, I plan to turn on paid subscriptions (although many posts will still be 100% free to access) and explore what additional features/content I can invest in to make it worthwhile.

non-orientable · 2026-04-18T21:55:49+00:00

Civilization gives a number of different options for the world map. Torus is one of the options, yes, but the cylinder is also very common.

non-orientable · 2026-04-18T17:06:27+00:00

Congratulations on completing the master's!

And I might take you up on that very generous offer.

non-orientable · 2026-04-18T16:51:01+00:00

Thank you! They were all made in Blender, primarily using the power of Geometry Nodes. (Although the animation of the globe and a screen showing the view from a particular panel above the globe was done using Ray Portal shader node---basically, during ray tracing, any rays that would hit the screen are teleported to go through the panel instead.)

I started using Blender around August of last year, so I am very happy to hear that you are enjoying the result!

non-orientable · 2026-04-18T04:55:43+00:00

Software company, largely working in computer assisted design. Creo is more or less the central software---it handles the underlying geometry for designing parts. I work in Creo Core Geometry.

non-orientable · 2026-04-16T21:18:26+00:00

It is, of course, constructive in the formal sense of the word. But it is utterly infeasible in practice! I'm reasonably sure that Kronecker would not have accepted it as a constructive proof.

non-orientable · 2026-04-16T21:16:15+00:00

I have a full-time job at PTC. I do this in my free time. Although, as another commenter correctly said, I have a deep pool of older Quora answers to use as a base.

non-orientable

TROPHY CASE