Jason Sachs jason-s

[Claude 4.6 Sonnet, May 1 2026]

[Jason] For 2048-bit odd numbers, what fraction of them are multiples of at least one of the first 256 prime numbers?

[Claude] This is a classic inclusion-exclusion / sieve problem. Let me work through it carefully.

Setup

We want: among all 2048-bit odd numbers, what fraction are divisible by at least one prime in {3, 5, 7, 11, ..., p₂₅₆} where p₂₅₆ = 1619?

[Jason] Please factor $x^4+4x^3+18x^2+28x+77$

[ChatGPT] Try factoring it as a product of two quadratics. A good guess is something symmetric like

$$(x^2 + 2x + a)(x^2 + 2x + b).$$

Expanding gives:

$$x^4 + 4x^3 + (a+b+4)x^2 + (2a+2b)x + ab$$

For 2048-bit odd numbers, what fraction of them are multiples of at least one of the first 256 prime numbers?

LaTeX	(render)	typst
`$$\sum_{i=0}^n i^2 = \frac{(n^2+n)(2n+1)}{6}$$`	$$\sum_{i=0}^n i^2 = \frac{(n^2+n)(2n+1)}{6}$$	`sum_(i=0)^n i^2 = ((n^2+n)(2n+1))/6`
`\lim_{\Delta x \to 0} \frac{\sin (x + \Delta x) - \sin x}{\Delta x} = \frac{d}{dx} \sin x = \cos x`	$$\lim_{\Delta x \to 0} \frac{\sin (x + \Delta x) - \sin x}{\Delta x} = \frac{d}{dx} \sin x = \cos x$$	`lim_(Delta x->0) (sin (x + Delta x) - sin(x))/(Delta x) = d/(d x) sin x = cos x`
`\mathbf{T}(\theta) = \begin{bmatrix}\cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{bmatrix}`	(won't render in this form in GitHub Markdown math in a table)	`#let bf(x) = $upright(bold(#x)) #set math.mat(delim: "[") bf(T)(theta) = mat(cos theta, -sin theta; sin theta, cos theta)`

This replaces MathJax inline markup in an <svg> element with SVG-rendered MathJax.

Usage is easy:

<html>
    <head>

	#set page(width: 20cm, height:auto)
	#set heading(numbering: "1.")

	= Fun with typst

	The typst program is pretty neat!

	- Extremely fast incremental recompile
	- Easier syntax
	- Markup includes a well-designed scripting language

	<!DOCTYPE html>
	<html>
	<head>
	<style type="text/css">
	body {
	font-family: 'Segoe UI', 'Lucida Grande'
	}
	table {
	border: 1px solid black;
	border-collapse: collapse;