Terminology: elegant proofs What do mathematicians mean when they say: that's an "elegant proof" of such and such. What are the ingredients of an elegant proof? Maybe you can give examples of elegant proofs of your own.
 A: Of the dictionary definitions, I think the one that most applies for "elegance" is 

Dignified gracefulness or restrained beauty of style.

It is an aesthetic judgement, but thing that cause me to consider a proof elegant are


*

*Brevity

*Simplicity - does the proof avoid a lot of case-by-case analysis

*Edifying - the proof is not just accurate, but emotionally feels convincing. Sometimes, you read a proof, and you understand it, but it feels more like an "accident," like why the fourth digit of $\pi$ is $1$.

A: For a collection of outstanding examples, Aigner and Ziegler's "Proofs from THE BOOK". Others are Dunham's books "Journey through Genius: The Great Theorems of Mathematics" and "The Mathematical Universe: An Alphabetical Journey Through the Great Proofs, Problems, and Personalities".
A: An elegant proof is a proof that makes everything much simpler than previously thought to be and usually provides insight and is very clear. Examples of elegant proofs are the following:
Euclid's proof that there are infinite prime numbers
Proof that a system of linear equations over the reals can have 0,1 or infinite solutions
Proof that there is only one identity in a group or only one inverse for every element in a group.
