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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.

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marked as duplicate by Asaf Karagila, Henry T. Horton, 5pm, Paul, Ittay Weiss Mar 1 '13 at 2:13

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up vote 2 down vote accepted

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".

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I like your post :) – saadtaame Mar 1 '13 at 0:35

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 $1$.
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I like that Edifying thing :D – saadtaame Feb 28 '13 at 23:52

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.

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Your second bit isn't quite correct. For example, over the integers mod $4$, $2x=2$ has two solutions, $x=1, 3$. A good salvage is to look at linear equations over a field. However, over a finite field your statement becomes correct, as long as we change "infinite" to mean "every element of the field". For example $x+1=x+1$ has $p$ solutions over $\mathbb F_{p}$. You might be better of to changing it to: "Nonzero polynomials over a field have $0$ or $1$ roots, but I'm not sure that's very elegant. – Dylan Yott Feb 28 '13 at 23:48
uhm can we just make it over the reals? I'm in high school – Carry on Smiling Feb 28 '13 at 23:54

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