Zsigmondy’s theorem is a powerful result about the prime divisors of $a^n-b^n$, and can be used to solve a variety of math olympiad problems (see for instance this blog post by KingSmasher3). In this post, I will present an elementary proof of Zsigmondy’s theorem.
I have just finished the final presentation for my undergrad thesis, titled Bicuspidal Geodesics on Punctured Hyperbolic Surfaces. (The slides are available on my portfolio page.) As such, I’m taking this opportunity to briefly describe my project.
Having been recently offered admission into some amazing math graduate programmes, I just returned from a two-week tour of some great schools all over the US.
Here is a list of advice (or warnings!) for incoming grad students, from current grad students and faculty.
Recently, I had to use a classical number theory estimate for my undergrad thesis. By pure coincidence, I ran into my batchmates Jessica and James revising for an analytic number theory class, and they taught me exactly the method I needed. In this post, we go through the Dirichlet hyperbola method for estimating certain sums of arithmetic functions.
In this note, we give a quick overview of the core ideas of Markov chain Monte Carlo methods, used to sample from probability distributions.
The week after Christmas is a great time to think about the coming year, and how to make it better than the last. I am not the kind of guy to make resolutions, but recently I have had a few ideas on how to improve myself, so I’ll try these out and see if anything sticks.
I recently read this mind-bending blog post by Joel David Hamkins that proves this result: there is a Turing machine program $P$ such that for any function $f:\bb N\to\bb N$—possibly uncomputable!—there is a model of Peano arithmetic (PA) in which $P$ computes $f$ on the standard natural numbers. If this statement doesn’t surprise you, I’m not sure what else would (except maybe those who work in logic or set theory; I’m convinced that these 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 informally explain (to myself, at least) how such a theory can be constructed, and why it is not a contradiction in terms.
Last week, I learnt about Micro Mages, a new game for the NES from Morphcat Studios. The game looks great, sounds great, and fits into 40KB. This video clearly explains the ideas behind the code and graphics optimisation.
Today I came up with a new idea to prove the famous identity
There is no shortage of existing proofs of this identity; see this math.SE question, or this collection of proofs compiled by Robin Chapman. I was inspired by this illuminating video, where 3Blue1Brown explains an elementary proof based on Euclidean geometry, by a physical interpretation involving systems of lighthouses. My new idea comes from trying to translate this proof into the complex plane.
Over the Simcoe Day long weekend, I got back into one of my (many) nerdy interests: pen-and-paper calculation.
No, not basic arithmetic (${+}{-}{\times}{\div}$); anyone can multiply two 10-digit numbers together, if they’re patient (and careful!) enough. I’m talking about the funky buttons on the calculator, like $\sqrt[n]{\quad}$, $\ln$, $\exp$, $\sin$, $\pi$, … They never taught you how to do those in school, did they?
In this series of posts, I’ll be exploring different ways of calculating mathematical functions and constants by hand.