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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? – Shivam Sarodia Mar 4 '14 at 23:00
It took a surprisingly long time (17th century) for even very basic concepts of probability theory to be developed, considering that those concepts would have been immensely valuable in real life, given that gambling has been popular forever. – dfan May 1 '14 at 21:57

34 Answers 34

James A. Garfield (20th President of the United States) found 1875 (~2175 years after Euclid) a quite simple, yet clever trapezoid proof of the Pythagorean theorem. Here is a video explaining it:

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The one that bumps up in my mind is Erdos-Mordell inequality. It states that for a point $ p $ in a triangle $ ABC $, with $\alpha ,\beta,\gamma $ denoting the distance of $ P $ to the 3 sides,respectively, then $ AP+BP+CP\geq 2 (\alpha + \beta +\gamma)$ This wasn't thought of until Erdos conjectured it in 1935.

P.s.People think Erdos might have been inspired by Euler's theorem for triangles, which itself is a 'shame' of Euclid.

P.s.2 I can't easily type the proper Hungarian letter on my phone, so sorry.

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The Meyers Serrin theorem that $ H=W $ (see their paper with exactly this equality as title) came only decades after people had proven all kinds of things with both $ H $ and $ W $, and is so elementary that you can ask second year students to prove it as exercise (extra credit, to be fair)

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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 '14 at 18:54
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 '14 at 19:07
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 '14 at 12:47
3… – oakad Mar 6 '14 at 5:50
The proof depends on Ribet's proof of the $\epsilon$-conjecture and Wiles' proof of the Taniyama-Shimura-Weil conjecture. Neither is trivial even to state, let alone prove. – anomaly Nov 10 '14 at 5:43

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