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On wikipedia there is a claim that the Abel–Ruffini theorem was nearly proved by Paolo Ruffini, and that his proof spanned over $500$ pages, is this really true? I don't really know much abstract algebra, and I know that the length of a paper will vary due to the size of the font, but what could possibly take $500$ pages to explain? Did he have to introduce a new subject part way through the paper or what? It also says Niels Henrik Abel published a proof that required just six pages, how can someone jump from $500$ pages to $6$?

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This question is like poetry. I love it! – user58514 Apr 21 '13 at 2:14
Sadly, only somebody with the stamina to slough through 500 pages of proof will be able to answer... – vonbrand Apr 21 '13 at 2:17
As Ruffini never said (but should have), "I have discovered a truly marvelous proof of this, which this 500 page book is too narrow to contain." – Michael Joyce Apr 21 '13 at 2:35
I like this question, but I dislike the answers. Not that they're poorly written, but I feel they don't really address the main point of this question. I knew before about a lot of these long proofs (including Mochizuki's, classifcation of finite simple groups, Feit-Thompson) but I like this question because I really want to know about Ruffini's proof! – user5501 Apr 21 '13 at 2:45
The relevant text in the wikipedia article is ambiguous to me: "While Cauchy felt that the assumption was minor, most historians believe that the proof was not complete until Abel proved this assumption. The theorem is thus generally credited to Niels Henrik Abel, who published a proof that required just six pages in 1824." Does the six page proof pertain to that assumption only? – Karl Kronenfeld Apr 21 '13 at 3:05

Not only true, but not unique. The abc conjecture has a recent (2012) proposed proof by Shinichi Mochizuki that spans over 500 pages, over 4 papers. The record is the classification of finite simple groups which consists of tens of thousands of pages, over hundreds of papers. Very few people have read all of them, although the result is important and used frequently.

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very few? no one! – user88576 Jan 3 '14 at 0:53
Not so, see this pdf, written by Mochizuki himself. Five people are named, who each went through the entire work in great detail. – vadim123 Jan 3 '14 at 1:44

Math can be very difficult.

There are famous long proofs, like that of the Feit-Thompson theorem, whose initial proof took 255 pages of very intrincate arguments, or the classification of quasi-thin simple groups done by Aschbacher and Smith in 1221 pages, or well, the whole classification of simple finite groups —which is estimated in the tens of thousands of pages— of which the result of Aschbacher and Smith is a very small part.

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