# Proof Gallery

Mathematics (like programming) is often called an art form. This makes mathematics the art of argument, not in the sense that anything is debatable, but in the sense that the beauty of a proof lies in its method. Analogously, a painting is not necessarily beautiful only for what it depicts or the medium used, but rather the way aspects like brushstroke, inspiration, and contemporary social attitudes combine in the artist’s self-expression. In mathematics as well, an aesthetic proof is an elegant, inspired, concise, and beautiful expression. Here we present is a collection of aesthetic proofs of more or less simple facts we’ve come across over the years. As time goes on, we may add more advanced proofs. For now, we’ll stick to common problems, pushing some well known solutions to the next level, and providing some proofs of our own discovery.

## Number Theory & Combinatorics

Sums of k powers
Sum of the first n numbers, sum of the first n squares
An arithmetic expression for $\binom{n}{2}$
There are infinitely many primes (a lower bound on $\pi(n)$)
Ramsey number lower bound $R(m,m)$
Learning a single variable polynomial
A parlor trick for SET

## Logic

n-Colorability is equivalent to finite n-colorability

## Analysis

Cauchy-Schwarz inequality (by amplification)

## Abstract Algebra

The smallest non-cyclic simple group has order 60
$\mathbb{Z}[\sqrt{2}]$ has infinitely many units

## Topology

There are infinitely many primes