Yosemite 小游记
几个星期前，一位小伙伴 wzc 问我有啥娱乐活动。聊着聊着，我们就约起了 Yosemite。于是几天前，我们终于趁着还不需要预约的时候，赶个 ddl 去了趟 Yosemite.
TL;DR 实在是妙啊！不愧是国家公园啊！瀑布量大管饱。
Changelog - 网站小更新
我最近又准备小翻新一下博客了。虽然东西没写多少，但是主题更新的快呀哈哈哈哈哈毕竟差生文具多
Sum-free subset 老题
一个集合 \(S\) 被称为 sum-free set 当且仅当 \(\forall a, b \in S, a + b \not\in S\)。
试证明：对于任何一个集合 \(A \subseteq \ZZ\backslash\{0\}\)，其最大的 sum-free subset 大小至少为 \(|A| / 3\).
（有一个和题面差不多长度的解答）
解答
对于 \(t \sim U[0, 1]\)，令 \(A_t := \{x \in A: 1/3 < xt \bmod 1 < 2/3 \}\)。显然 \(A_t\) 为 sum-free set. 最后再注意到 \(\max_t |A_t| \geq \mathbb{E}_t [|A_t|] = |A|/3\) 即可。
解法来自 Erdos。
Ten Days in the Zed Editor
Recently I'm playing with a new code editor, Zed. I used it for about ten days and here write a review to share my experiences with it. This review is based on the latest version (v0.186.9).
TL;DR
My feelings about Zed are mixed: as an editor, it's pretty good; as a (Python) code editor, it's pretty bad; as an AI code editor, it's reasonable. As of its current state (v0.186.9), it's hard for me to recommend it to colleagues due to its poor Python experience, but I think I'll keep an eye on it.
Update (@2025-05-19): After tuning LSP configurations, the Python experience is much better. See below.
Integer and Rational Solutions of a Binary Quadratic Equation (0): General binary quadratic equation
In this post, we are interested in finding all integer and rational solutions for a general binary quadratic equation: \[ \begin{equation} ax^2 + bxy + cy^2 + dx + ey + f = 0, \label{eq:general} \end{equation} \] for any integer coefficients \(a, b, c, d, e, f\). The equation might degenerate; we only point it out when it does but do not solve it, as it's typically easier to solve a degenerate equation.
This post mainly serves as an entry point for a series of posts and also contains a little bit of the story behind the posts. That is why this post has index 0 in the title.
Integer and Rational Solutions of a Binary Quadratic Equation (5): Binary quadratic form
In this post, we are interested in finding all integer solutions of the following equation: \[ \begin{equation} ax^2 + bxy + cy^2 = n. %\label{eq:ibqf} \end{equation} \] where \(acn \neq 0\), \(b^2 - 4ac \neq 0\). This equation can certainly be solved using methods from the generalized Pell equation, but here, we explore other methods.
Integer and Rational Solutions of a Binary Quadratic Equation (4): Pell equation and generalized Pell equation
In this post, we are interested in finding all integer solutions of Pell equation \[ \begin{equation} x^2 - d y^2 = 1, \label{eq:pell} % \tag{Pell Equation} \end{equation} \] and generalized Pell equation \[ \begin{equation} x^2 - d y^2 = n, \label{eq:generalized-pell} % \tag{Generalized Pell Equation} \end{equation} \] for a non-square integer \(d > 0\) and \(n \neq 0\).
Integer and Rational Solutions of a Binary Quadratic Equation (3): $ax^2+by^2=n$
In this post, we're interested in finding all integer solutions of a binary quadratic equation: \[ \begin{equation} ax^2 + by^2 = n, \label{eq:ellipse} \end{equation} \] for \(a, b, n > 0\).
Integer and Rational Solutions of a Binary Quadratic Equation (2): Quadratic Residue
In this post, we're interested in finding any or all integer solutions of a binary quadratic equation: \[ x^2 - ny - a = 0. \] We assume that the equation is non-degenerate in the sense that \(n \neq 0\).
Integer and Rational Solutions to a Binary Quadratic Equation (1): Legendre equation
In this post, we're interested in finding any or all rational solutions of the following equation: \[ ax^2 + by^2 + c = 0, \] or alternatively, all integer solutions to the Legendre equation: \[ \begin{equation} ax^2 + by^2 + cz^2 = 0, \label{eq:legendre} % \tag{Legendre Equation} \end{equation} \] where \(abc \neq 0\).