SIMO Retiree Blog

(Xu Chen here) Today I am going to talk about a subtle but important thought process for many problems, which is how you visualize the objects you are working with. This is more of an intuitive idea than rigorous math, so forgive me if not everything is super clear in this post. How you look at the problem can heavily influence what ideas you can come up with: The solution often follows from a good understanding of the underlying structure. Example 1. Let's start with a simple example. (USAJMO 2019/2) Let $\Z$ be the set of all integers. Find all pairs of integers $(a,b)$ for which there exist functions $f:\Z\to\Z$ and $g:\Z\to\Z$ satisfying $$f(g(x))=x+a\quad \text{and}\quad g(f(x))=x+b$$ for all integers $x$. Hopefully it is clear that we cannot treat this as a standard FE, but must instead consider some combinatorial ideas. Let's think of $\Z$ on the number line. We draw red arrows to represent $x\mapsto f(x)$ and blue arrows to represent $x\mapsto g(x)$: One cond...

(This is Yan Sheng.) $\newcommand{\bb}{\mathbb}\newcommand{\Con}{\operatorname{Con}}$I once read this mind-bending blog post by Joel David Hamkins that proves this result: Theorem : There is a Turing machine program $P$ such that for any function $f:\bb N\to\bb N$, there is a model of Peano arithmetic in which $P$ computes $f$ on the standard natural numbers. Read that again: there is a single Turing machine that outputs any given function—even uncomputable ones (!!)—if you run it in the correct universe. If this statement doesn’t surprise you, I’m not sure what else would (unless you work in logic or set theory; I’m convinced that those people routinely believe as many as six impossible things before breakfast ). A extraordinary claim needs an extraordinary proof, and when I read through Hamkins’s argument I came across a lesser absurdity: a theory that proves its own inconsistency. Since all my knowledge of formal logic is from folklore, this post is my attempt to informal...

(David here.) In The Rising Sea , Ravi Vakil writes: When introduced to a new idea, always ask why you should care. Do not expect an answer right away, but demand an answer eventually. Try at least to apply any new abstraction to some concrete example you can understand well. We have something analogous for understanding proofs - with time, new perspectives show up and reveal new insight, and allows one to tackle problems that seemed impossible before.

(Zhao Yu here.) It's been a long time since I've touched olympiads and I have a really bad memory, so today I'll try something different and talk about something unrelated to olympiads. I will talk about 3 interesting facts that follow from the theory of elliptic curves . I will not be precise, and I will black-box many things. The point is just to give you a taste of elliptic curves and hopefully this will motivate you to learn the actual theory in the future!

 (Dylan here.) For this post (and perhaps my subsequent blog posts), I will share a math-adjacent slice of my experience, and then talk some (probably unrelated) math. I will also include upfront some problems related to the non-math/math I am talking about, in case you want to try them first.

(Jit here). Today I am going to write about the Pell's equation $x^2 - dy^2 = 1$, which hopefully should be familiar to readers, and about generalizations of it to higher degrees. Let's first review some facts about the integer solutions of a Pell's equation.