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Is there a canonical example of a simple/short theorem which has a very complex/long history?

I'm giving a presentation and would like to illustrate the point that presenting only the final result hides the interesting work and history behind the result.

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  • $\begingroup$ Fermat's Last Marginalium? $\endgroup$ – Chappers Mar 16 '17 at 22:46
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    $\begingroup$ The Parallel Postulate? $\endgroup$ – lulu Mar 16 '17 at 22:57
  • $\begingroup$ Should the theorem just be simple/short to state, or does it need a similarly simple/short proof? $\endgroup$ – pjs36 Mar 16 '17 at 23:38
  • $\begingroup$ @pjs36 both would be best. $\endgroup$ – lowndrul Mar 19 '17 at 0:04
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  • The simple concept of a function has a rich and long history. It is Dedekind, in 1888, that first formalised its general definition.

  • The Pythagorean theorem was known to egyptians and mesopotamians, but was only shown in a mathematical framework by the greeks. It is related to the existance of irrational numbers, which stirred much trouble among the pythagoricians.

  • The four color theorem. It states that, using four colors, any planar map can be colored so that no two adjacent regions have the same color. It is hard to prove. So difficult, in fact, that computers are used to write and verify the proofs. This raises interesting questions about what it means to prove a mathematical theorem.

  • Squaring the circle. The problem, thousands of years old, is to construct a square, using ruler and compass, that has the same area as the circle. It was shown to be impossible in 1882, but many mathematical cranks (see this book) are still trying to solve it!

Some interesting recent results (there's a lot of history but you'll have to dig):

  • How can you calculate the $n$th digit of $\pi$? A solution was given in 1995 by Simon Plouffe.

  • How many times should you shuffle a deck of card for it to be well mixed? The answer was given in 1992 by Bayer and Diaconis here.

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  • $\begingroup$ Worth noting Plouffe's formula doesn't give us $n$th decimal digit. $\endgroup$ – Wojowu Mar 17 '17 at 8:00
  • $\begingroup$ Touche, I thought you were refering to the original BPP formula. $\endgroup$ – Wojowu Mar 17 '17 at 11:32
  • $\begingroup$ Thanks for pointing this out. His method can also be used to calculate the decimal digits, but this takes more time. $\endgroup$ – Olivier Mar 17 '17 at 11:44
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I would go for the solution to the cubic equation and all the very real, at times very petty and ultimately quite sad for many of the participants, human drama involving the who-came-up-with-it-first question.

For more an internet search will do the trick, but a brief history can be found here or here for example.

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A canonical example of such a result that has both geometric appeal and a rich history is Euler's theorem that a polyhedron satisfies $V-E+F=2$ where $V,E,F$ are respectively the numbers of faces, edges, and vertices of the polyhedron. Its rich history includes a number of incomplete proofs and further generalisations. Today the result is a fundamental theorem at the center of much research in algebraic topology. The theorem attracted the attention of philosopher Imre Lakatos whose thesis "Proofs and Refutations" examines the history of the theorem. See this link.

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