## A Tea-Time Paper Post

During our (now extended) time cooped up at home I thought it would be fun to put together a small post consisting of links to some very interesting relatively short papers that can be read with your favorite cup of tea. Descriptions are given before each link with the necessary background. Stay safe everyone!

#### Golden Product Identity for $e$

Euler’s identity $e^{i\pi}+1 = 0$ is a famous result of an interesting relationship between important mathematical constants. This short paper is a “big brother” result giving a relationship between $e$ and three important number theoritic tools. Only a course in analysis is necessary to read the paper; see here.

#### A Simple Proof That $\pi$ Is Irrational

Ever wanted to see the proof for why $\pi$ is irrational? This proof is less than a page and requires only basis calculus; see here.

#### A One-Sentence Proof That Every Prime $p \equiv 1 \pmod{4}$ Is a Sum of Two Squares

While not a constructive proof, this argument is the shortest one in the literature. It exploits a involution between triplets of integers and needs very little background; see here.

#### A Proof of Liouville’s Theorem

Just to be clear, this is a proof of Liouville’s theorem on harmonic functions. It’s a geometric argument and only the basics of harmonic functions and spheres are necessary; see here.

#### Most Infinitely Differentiable Functions are Nowhere Analytic

This is quite an interesing result. It says the set of real-valued smooth functions on $\mathbb{R}$ that are analytic at some point are a very small class of all smooth functions. In particular, they form a first category subset. This means the subset can be written as a countable unions of nonwhere dense subsets. A basic understanding of analysis is all that’s needed for the proof; see here.

#### On the Uniqueness of the Cyclic Group of Order $n$

By Lagrange’s theroem it’s immediate that $\mathbb{Z}/p\mathbb{Z}$ is the unique cyclic group of order $p$ for $p$ prime. A natural question to ask is if this happens more generally. The answer is yes, when $(n,\phi(n)) = 1$ where $\phi(n)$ is Euler’s totient function. Only basic group theory is used in the argument; see here.

#### On Manifolds Homeomorphic to the $7$-Sphere

This is a famous paper by J. Milnor where he explicitly constructes exotic spheres of dimension $7$. Unfortunatly, this paper requires a standard background in smooth manifold theory and algebraic topology. See here for the paper.

#### Recounting the Rationals

Everyone who has taken a first course in proofs has seen the argument for why the rationals $\mathbb{Q}$ are countable. This is usually described by moving diagionally along an array. This paper explicitely constructs a sequence which couts every rational number once. Little to no background is required; see here.