Identity wrote:JaneFairfax wrote:Is this the most difficult proof ever?

No. That would probably be, "Classify all finite simple groups." That spanned thousands of papers and hundreds of mathematicians. Another good one is "Prove that any group of odd order is solvable." That one was 255 pages.

I believe we have a new contender for the record:

Baffling ABC maths proof now has impenetrable 300-page ‘summary’.

A summary of a massive mathematical proof that has baffled mathematicians for the past five years may help a few more people get to get grips with the key ideas. How long is the explainer? A mere 300 pages.

And that is only the summary: the original work – Shinichi Mochizuki’s proof of the ABC conjecture published in 2012, using a radical new theory developed over two decades – contained over 500 pages.

]]>mathsyperson wrote:What about the 4-colour proof?

The one that says it only ever takes 4 colours to fill in a map of regions such that no two regions sharing a border (points don't count) have the same colour.Does anyone know how long that one is, and if it compares to some of the others here?

What I meant is in terms of conceptual and mathematical difficulty. The 4-colour theorem (I think) is a computer proof, and covers lots of cases. This may make it long, but it may not make it difficult.

Ah, so you did. Ricky distracted me by mentioning the length of his example, but that one's probably long *and* difficult.

You should listen to my warning. Go to college, get an undergraduate degree in mathematics, then a few years in graduate school. Then you can start trying.

Sry i didnt take your warning seriouly the first time but after reading the first 2 pages i think i shall.

]]>Thx Jane that was very useful. Thats gonna occupy me for an year or so. I m goin to try and understand it if i can.

You should listen to my warning. Go to college, get an undergraduate degree in mathematics, then a few years in graduate school. Then you can start trying.

]]>What about the 4-colour proof?

The one that says it only ever takes 4 colours to fill in a map of regions such that no two regions sharing a border (points don't count) have the same colour.Does anyone know how long that one is, and if it compares to some of the others here?

What I meant is in terms of conceptual and mathematical difficulty. The 4-colour theorem (I think) is a computer proof, and covers lots of cases. This may make it long, but it may not make it difficult.

]]>It's the longest (definitive) problem that was every constantly worked on.

]]>The one that says it only ever takes 4 colours to fill in a map of regions such that no two regions sharing a border (points don't count) have the same colour.

Does anyone know how long that one is, and if it compares to some of the others here?

]]>Identity wrote:JaneFairfax wrote:Is this the most difficult proof ever?

No. That would probably be, "Classify all finite simple groups." That spanned thousands of papers and hundreds of mathematicians. Another good one is "Prove that any group of odd order is solvable." That one was 255 pages.

Perhaps it's not the longest proof, but has any other theorem taken longer to prove from the time it was first proposed? My hunch is that there has been, but I don't know off the top of my head.

]]>JaneFairfax wrote:Is this the most difficult proof ever?

No. That would probably be, "Classify all finite simple groups." That spanned thousands of papers and hundreds of mathematicians. Another good one is "Prove that any group of odd order is solvable." That one was 255 pages.

]]>Is this the most difficult proof ever?

]]>I am interested to know if the theorem was valid even before it was proved?

Unless you think there were integers that fit the theorem before Wile's proof and disappeared as soon as he came up with it, yes it was valid.

Is there any corollary to the theorem?

Not quite. Rather, Fermat's Last Theorem is a corollary to a rather huge theorem. That rather huge theorem is what Wiles proved, known as the Modularity theorem. Specifically, Wiles proved that all elliptic curves were modular, and if there was a solution to FLT, then it would be an elliptic curve that wasn't modular.

FLT is simply a single example of the theorem he proved.

Which other problems make use of Fermat's Last theorem??

None that I'm aware of.

Some people think that Fermat's proof had a flaw in it (there are certainly some flaws that could have come up, as people who tried proving it have shown).

I think he was just making it all up...

]]>After he proposed this theorem, he later proved that no integers a, b and c exist such that a^4 + b^4 = c^4. If he believed in his theorem, then there would be no need to do that, so people think he made a mistake and then discovered it later.

]]>Fermat was right when he wrote

I am interested to know if the theorem was valid even before it was proved?

Is there any corollary to the theorem?

Which other problems make use of Fermat's Last theorem??

I believe it was the EASIEST problem for anyone TO UNDERSTAND ((in contrast, Reimann's Hypothesis is something which needs a good deal of knowledge of Set Theory)) and yet to be the TOUGHEST problem TO SOLVE (till '94)!!

http://math.stanford.edu/~lekheng/flt/wiles.pdf

Happy reading!

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