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 $6$?
[Math] On a long proof
math-historyproof-writing
Related Solutions
Although Riemann was the first to prove the functional equation for complex $s$, the ideas certainly predate Riemann. Euler knew something equivalent to the functional equation for integer values of $s$, via Abel summation where the series does not converge. Weil's 1975 paper "Two lectures on number theory, past and present" (in his collected works vol. 3) says that before Riemann, both Schlomilch and Malmquist published the functional equation for the Dirichlet $L$-series attached to the nontrivial character modulo $4$. This paper is a good place to begin for thinking about how Riemann's ideas evolved.
For $3$ or $4$ digit numbers you were using $$(10a+b)^2=100a^2+(20a+b)b$$
For $5$ or $6$ digit numbers you would have used the messier $$(100a+10b+c)^2=10000a^2+(2000a+ 100b)b+(200a+ 20 b+c)c$$
So you would separate the number to be square-rooted into $100$s, in your example $27,04$
and it looked like
? ?
------
|27,04
The largest square less than or equal to $27$ is $25=5^2$ so you write
5 ?
------
5 |27,04
|25
---
| 2,04
You then double the $5$ you have written at the top to give $10$ and then add an extra digit $X$ so $10X \times X$ is a large as possible but does not exceed $204$: $102 \times 2= 204$ works exactly
5 2
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5 |27,04
|25
---
102 | 2,04
| 2,04
------
0
So $\sqrt{2704} =52$
If the result had not been exact, you could have continued the same way, bringing down two more digits (possibly $00$).
My school thought this a waste of time and instead taught us to use logarithm tables.
Best Answer
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.