SIMO Retiree Blog

(This is Glen.) I was sorting through the LaTeX files on my computer and unearthed an old set of solutions to a mysterious problem set, dated 2020. After a bit of digging in some old Discord servers, I found out that these were solutions to one of Zhao Yu's sets for some RI training, which I had presumably crashed because it was online (thanks to Covid) and I was too free or something. Anyway, this file contained a lengthy introduction to UFDs, which I had recently learnt about in uni and had used to overkill a couple of problems in the set. This is, I think, quite suitable for a blog post, so here we are. The fundamental theorem of arithmetic As a warmup, let's think about something we learn about in primary school (well, at least I remember learning about this in primary school but I am old so this may no longer be the case): the unique prime factorisation of integers. (Fundamental theorem of arithmetic) Each integer $n>1$ can be written uniquely as $n=p_1\cdots p_k$, wher...

 (Zhao Yu here). Today someone spoke about this short proof of a lower bound of the number of primes, which I found it too nice to not share.

(David here.) We've had at least two articles on elliptic curves so far: Zhao Yu's article talks about some neat applications across various areas of math, while Dylan's article focuses on motivating the group law. Recently, I found out that the group law can be used to tackle actual Olympiad geometry problems (!), so I thought I might write a guide as to how.

(David here.) Recently a problem showed up on the problems feed in the SIMO Retirees' Discord server that made me take a second look at Desargues' Involution Theorem, and I realized it was actually the perfect starting point to discuss the interplay between geometry and algebra!