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I recently heard about Eric Lu’s announcement that RSA-260 had been factored. Congratulations to Eric!

This is a bit of a non-post, minimally edited stream-of-conciousness. If that bothers you, no need to read it. I just wanted to write the following down somewhere.

This exact factorization is a project I had been working on, on-and-off, since mid-2023. Since November 2025, I had begun putting in serious effort. I don’t have institutional resources, so all work was self-funded, chugging away slowly. I was about one month away from finishing relation collection when I got wind of Eric’s work.

This post motivates the complete zeta function $\zeta_{\mathbb{A}}(z) = \pi^{-z/2} \, \Gamma(z/2) \, \zeta(z)$, explaining how it arises in a natural way by including contributions from all possible completions of $\mathbb{Q}$. The argument is from Andreas Knauf’s excellent lecture notes; I just fill in some details.

This post documents the implementation of the error function $\mathrm{erf}(z)$ used in my complex function plotter. For small $z$, I use the asympotic series by Abramowitz & Stegun. For large $z$, I use a custom expansion around the 45° line.

I recently found a paper titled ‘A Cone Eversion,’ by S. Tabachnikov. I found it a bit difficult to see what was happening in 3D, so I made a visualization. Enjoy!

Inspired by Matt Parker’s recent video, I decided to search for primes $p$ with $\tan{p} > p$, the first of which is the 46-digit

$$p = 1169809367327212570704813632106852886389036911.$$

$$p = 116980936732 \dots 886389036911.$$

How do you go about finding more?

Well, we want $\tan{p}$ to be big — very big. From high-school trigonometry, we know that this occurs when $p$ is just a tiny bit less than a half-integer multiple of $\pi$. In other words, we want

One possibility is to write a fragment shader that repeatedly downscales the image by a factor of 2, where each output pixel is the average of the four pixels in its preimage. Assuming each render takes constant time due to GPU parallelization,^[1] this technique can average an $n \times n$ texture in $\log_2{n}$ time.