Beautiful theorems with short proof - MathOverflow [closed] most recent 30 from http://mathoverflow.net 2013-06-19T19:38:03Z http://mathoverflow.net/feeds/question/106859 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/106859/beautiful-theorems-with-short-proof Beautiful theorems with short proof Jörg Neunhäuserer 2012-09-10T23:55:37Z 2012-09-11T05:23:13Z <p>I like to ask for beautiful mathematical theorems with short proof. A proof is short in my sense if it is at most one page assuming basic notations and very basic results a second year student will know and understand. I do not like to discuss what beauty means...I have now written down a script with 107 such theorems (in my sense) and their proof and I have ideas for another 20. One the one hand I would be very thankful for theorems I have yet not considered. On the other hand I like to see if there is some consensus which theorems with a short proof are beautiful.</p> http://mathoverflow.net/questions/106859/beautiful-theorems-with-short-proof/106863#106863 Answer by dmdmdmdmdmd for Beautiful theorems with short proof dmdmdmdmdmd 2012-09-11T00:45:30Z 2012-09-11T00:45:30Z <p>Picard's little theorem is remarkable, and its one-line proof led Littlewood to remark that it would be the world's shortest Ph.D. thesis. </p> http://mathoverflow.net/questions/106859/beautiful-theorems-with-short-proof/106866#106866 Answer by paul Monsky for Beautiful theorems with short proof paul Monsky 2012-09-11T01:02:58Z 2012-09-11T01:02:58Z <p>Fermat's proof, by infinite descent, that there is no Pythagorean right triangle whose area is a square might qualify.</p> http://mathoverflow.net/questions/106859/beautiful-theorems-with-short-proof/106867#106867 Answer by Igor Rivin for Beautiful theorems with short proof Igor Rivin 2012-09-11T01:16:47Z 2012-09-11T01:16:47Z <p>The standard evaluation of $\int_{-\infty}^{\infty} \exp(-x^2) dx.$</p> http://mathoverflow.net/questions/106859/beautiful-theorems-with-short-proof/106869#106869 Answer by Igor Rivin for Beautiful theorems with short proof Igor Rivin 2012-09-11T01:26:08Z 2012-09-11T01:26:08Z <p>@Paul Monsky's proof of Monsky's theorem: a complete proof starting from nothing takes <a href="http://stanford.edu/~moorxu/misc/equiareal.pdf" rel="nofollow">two pages.</a> (doesn't quite meet the criteria, but what the heck).</p> http://mathoverflow.net/questions/106859/beautiful-theorems-with-short-proof/106870#106870 Answer by Robert Israel for Beautiful theorems with short proof Robert Israel 2012-09-11T01:26:27Z 2012-09-11T01:26:27Z <p>L.M. Kelly's proof of the Sylvester-Gallai theorem: in any configuration of $n$ points in the plane, not all on a line, there is a line containing exactly two of the points.</p> <p>See Aigner &amp; Ziegler, "Proofs from the Book", chapter 8.</p> http://mathoverflow.net/questions/106859/beautiful-theorems-with-short-proof/106871#106871 Answer by an12 for Beautiful theorems with short proof an12 2012-09-11T01:30:00Z 2012-09-11T01:30:00Z <p>Euler's formula $$\mathrm{e}^{i \theta} = \cos \left( \theta \right) + i \sin \left( \theta \right)$$ when considered as a theorem. From whatever angle you look at it, almost all the proofs are short and extremely beautiful.</p> http://mathoverflow.net/questions/106859/beautiful-theorems-with-short-proof/106872#106872 Answer by an12 for Beautiful theorems with short proof an12 2012-09-11T01:55:16Z 2012-09-11T01:55:16Z <p>Cantor's diagonal argument to prove that $\mathbb{R}$ is uncountable.</p> http://mathoverflow.net/questions/106859/beautiful-theorems-with-short-proof/106880#106880 Answer by Jiang for Beautiful theorems with short proof Jiang 2012-09-11T03:16:25Z 2012-09-11T03:16:25Z <p>The proof of Brouwer fixed point theorem by using fundamental group of $S^1$ is equal to $\mathbb{Z}$, while the fundamental group of $D^2$ is trivial.</p> http://mathoverflow.net/questions/106859/beautiful-theorems-with-short-proof/106885#106885 Answer by paul Monsky for Beautiful theorems with short proof paul Monsky 2012-09-11T05:23:13Z 2012-09-11T05:23:13Z <p>The proof(via the pigeon-hole principle--continued fractions would need too much preparation) that when D>0 is not a square then the "Pellian equation" xx-Dyy=1 has a non-trivial solution.</p>