Reddit mentions of Proofs from THE BOOK

Yes. In fact, certain theorems get re-proven using different methods as a kind of "math golf" or fun puzzle for mathematicians.

Although their contents will likely be over your head, look at some of the various proofs of the fundamental theorem of algebra, for example.

Similarly, if an important but hard-to-prove theorem "falls out" of some newly-developed mathematical abstraction, that's considered a sign that the abstraction is the "right one" (doubly so if the abstraction wasn't developed with a proof of the original theorem in mind).

For example, Brouwer's original proof of what is now called Brouwer's fixed-point theorem was somewhat cumbersome, requiring lots of calculation, consideration of special cases, and other necessary-but-unenlightening bookkeeping. Using the more modern language of homology, however, the proof becomes very straightforward.

One could say that a "simple" or "elegant" proof manages to isolate exactly those things which convey the essence of "what's going on" with the theorem and related concepts. At a purely formal level a proof is a proof is a proof, but in practice an elegant proof offers a more visceral resolution to the question of "Why should this be true?"

Most mathematicians will collect their favorite proofs of various theorems. You'll often here them say things like "Oh, have you ever seen so-and-so's proof of XYZ theorem?" It's a lot like music fans being excited about sharing covers or remixes ("Oh, did you hear DJ so-and-so's remix of XYZ song?"). There's a sociology paper in there somewhere.

You might be interested in Proofs from THE BOOK.

Your post has too little context/content for anyone to give you particularly relevant or specific advice. You should list what you know already and what you’re trying to learn. I find it’s easiest to research a new subject when I have a concrete problem I’m trying to solve.

But anyway, I’m going to assume you studied up through single variable calculus and are reasonably motivated to put some effort in with your reading. Here are some books which you might enjoy, depending on your interests. All should be reasonably accessible (to, say, a sharp and motivated undergraduate), but they’ll all take some work:

(in no particular order)
Gödel, Escher, Bach: An Eternal Golden Braid (wikipedia)
To Mock a Mockingbird (wikipedia)
Structure in Nature is a Strategy for Design
Geometry and the Imagination
Visual Group Theory (website)
The Little Schemer (website)
Visual Complex Analysis (website)
Nonlinear Dynamics and Chaos (website)
Music, a Mathematical Offering (website)
QED
Mathematics and its History
The Nature and Growth of Modern Mathematics
Proofs from THE BOOK (wikipedia)
Concrete Mathematics (website, wikipedia)
The Symmetries of Things
Quantum Computing Since Democritus (website)
Solid Shape
On Numbers and Games (wikipedia)
Street-Fighting Mathematics (website)

But also, you’ll probably get more useful response somewhere else, e.g. /r/learnmath. (On /r/math you’re likely to attract downvotes with a question like this.)

You might enjoy:
https://www.reddit.com/r/math/comments/2mkmk0/a_compilation_of_useful_free_online_math_resources/
https://www.reddit.com/r/mathbooks/top/?sort=top&t=all

Proofs from THE BOOK
It's "the bible" of the most elegant mathematical proofs, which Paul Erdös always talked about.

I wouldn't recommend reading research papers that early. The are usually awfully specific and tend to use incoherent notation.

If you want to read some nice proofs, check out Proofs from THE BOOK. It's a collection of beautiful proofs covering many topics.

Proofs from The Book

Proofs from THE BOOK. This is basically a showcase of mathematicians at their most clever. The book could be read by anyone with a solid grounding in calculus, linear algebra and discrete math.

• Putnam and Beyond is exactly what you're looking for. The difficulty of the problems in each chapter (including proofs) start at, I would say, "easy to tricky", depending on your strength, and quickly ascend to "Putnam level" and, as the name suggests, "Beyond Putnam level".
  
  The first chapters focus on specific methods of proof. Then the rest of the chapters focus on different areas of undergrad-level math. Each chapter has a few examples and useful tips, but it's mostly just problems. The second half of the book offers detailed solutions to each problem. At 814 pages, this book is bound to keep you entertained for a long time.
  
  
• Proofs from The Book is not quite what you're looking for, but is instead a collection of some of the most elegant or well known proofs in mathematics (mostly discrete maths) that are accessible to an undergrad. There are, for example, 6 different proofs to the infinity of primes. Also highly entertaining and is helping me a lot in seeing proofs in a different light, appreciating different styles and approaches.

As others have said, intelligence isn't everything. If you're willing to work hard, you can earn your bachelor's in mathematics.

But do you want to? What do you want to do with that degree?

Moreover, are you sure you really like math? College algebra and pre-calculus have very little in common with most math courses. At some point in a math curriculum, you'll be taking courses about abstract concepts that bear no obvious relation to the real world (unlike say Calculus and Differential Equations, in which real-world examples are abundant).

Furthermore, in those later classes, the question stops becoming: "What is the value of x?"^ Instead, those classes are more like: "Prove P(x) for all real numbers x". Proofs are different in kind from anything you've done so far in your math classes, and it will dominate all of the upper-level math courses you take.

Before you go down the path of majoring in mathematics, I recommend you get some exposure to proofs and try some on your own, to see if that's really something you're interested in. If your library has it, check out Proofs from THE BOOK, a collection of particularly beautiful proofs.

^ If you're good at solving equations and decide against majoring in mathematics, there are several other good fields to consider. Engineering and computer science, for example, offer great careers for the mathematically inclined.

https://www.amazon.com/Proofs-BOOK-Martin-Aigner/dp/3642008550

It's not exactly what you are looking for, but check out Proofs from THE BOOK.

http://www.amazon.com/Proofs-THE-BOOK-Martin-Aigner/dp/3642008550

I strongly recommend Proofs from THE BOOK.

Any suggestion for interesting proofs to remember?

I really love (one of) the proofs to Fermat's Little Theorem. I might actually be able to reproduce it.

There is of course Proofs from the book.

Proofs from The Book is a great collection of easy to understand and accessible proofs. As someone who majored in math, but who will be not pursuing mathematics at the graduate level for a while, I've enjoyed working through them.

It is in Proofs from THE BOOK, in section 25.3 on p. 163 of the 4th edition.

> Paul Erdős attributes the following nice application of the pigeon-hole principle to Andrew Vázsonyi and Marta Sved:

> Claim. Suppose we are given n integers a1, ..., an, which need not be distinct. Then there is always a set of consecutive numbers ak+1, ak+2, ..., aℓ whose sum Σi=k+1ℓ ai is a multiple of n.

Note the word consecutive; this is an even stronger claim.