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What are examples of mathematical results that were discovered surprisingly late in history? Maybe the result is a straightforward corollary of an established theorem, or maybe it's just so simple that it's surprising no one thought of it sooner.

The example that makes me ask is the 2011 paper John Baez mentioned called "Two semicircles fill half a circle", which proves a fairly simple geometrical fact similar to those that have been pondered for thousands of years.

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I'm likely off, but isn't the posted problem quickly solved by use of coordinate geometry? –  Draksis Mar 4 at 23:00

31 Answers 31

An example of this that jumps to my mind is Fermats last theorem, for which was conjectured by him in 1637 but only proved in 1995 despite many years work by countless mathematicians. But this proof (obviously) is not simple nor straightforward. I don't know how important that criteria is on our answers, but it is an example of a result that many believed to be true for a very long time, yet to be unable to be proved. Fermats last theorem is that,

For a,b and c being positive intergers

$a^n+b^n\neq c^n$ For n bigger than 2

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The proof is not straightforward, which is the importance condition in the question. If you remove this criteria then many more examples come to mind, like the impossibility of squaring the circle. –  Sawarnik Mar 4 at 18:54
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I don't think this satisfies the primary criterion, which is "surprisingly late". The theorem turned out to be surprisingly difficult, but there is no way that the proof we have could have been discovered by, say, Euler. –  MJD Mar 4 at 19:07
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Fermat allegedly had a proof, that was too big for him to put in the margin. It would have been much simpler than Wiles' proof. –  M. Knight Mar 5 at 12:47
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en.wikipedia.org/wiki/… –  oakad Mar 6 at 5:50

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