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hugo/content/posts/2013-06-26-sylow-theorems.md
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@@ -14,7 +14,7 @@ summary: "A fairly long and winding way through a proof of the three Sylow theor
---
(This post is mostly to set up a kind of structure for the website; in particular, to be the first in a series of posts summarising some mathematical results I stumble across.)
EDIT: There is now [an Anki deck](/AnkiDecks/SylowTheoremsProof.apkg) of this proof, and a [collection of poems][sylow sonnets] summarising it.
EDIT: There is now a [collection of poems][sylow sonnets] summarising this proof.
In Part IB of the Mathematical Tripos (that is, second-year material), there is a course called Groups, Rings and Modules. I took it in the academic year 2012-2013, when it was lectured by [Imre Leader](https://en.wikipedia.org/wiki/Imre_Leader). He told us that there were three main proofs of the [Sylow theorems](https://en.wikipedia.org/wiki/Sylow_theorems), two of which were horrible and one of which was nice; he presented the "nice" one. At the time, I thought this was the most beautiful proof of anything I'd ever seen, although other people have told me it's a disgusting proof.

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hugo/content/posts/2014-09-09-sum-of-two-squares-theorem.md
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@@ -12,7 +12,7 @@ aliases:
title: Sum-of-two-squares theorem
---
*Wherein I detail the most beautiful proof of a theorem I've ever seen, in a bite-size form suitable for an Anki deck. I attach the [Anki deck], which contains the bulleted lines of this post as flashcards.*
*Wherein I detail the most beautiful proof of a theorem I've ever seen, in a bite-size form suitable for an Anki deck.
# Statement
There's no particularly nice way to motivate this in this context, I'm afraid, so we'll just dive in. I have found this method extremely hard to motivate - a few of the steps are a glorious magic.
@@ -32,8 +32,6 @@ Additionally, we'll call a number which is the sum of two squares a **nice** num
## First implication: if primes 3 mod 4 appear only to even powers…
We prove the result first for the primes, and will then show that niceness is preserved on taking products.
* Let \\(p=2\\). Then \\(p\\) is trivially the sum of two squares: it is \\(1+1\\).
* Let \\(p\\) be 1 mod 4.
* Then modulo \\(p\\), we have \\(-1\\) is square.
@@ -81,5 +79,4 @@ That ends the proof. Its beauty lies in the way it regards sums of two squares a
[Gaussian integers]: https://en.wikipedia.org/wiki/Gaussian_integers
[UFD]: https://en.wikipedia.org/wiki/Unique_factorization_domain
[irreducible]: https://en.wikipedia.org/wiki/Irreducible_element
[prime]: https://en.wikipedia.org/wiki/Prime_element
[Anki deck]: {{< baseurl >}}AnkiDecks/SumOfTwoSquaresTheorem.apkg
[prime]: https://en.wikipedia.org/wiki/Prime_element