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Let's take an obvious example. I'm sure that amateur mathematicians (and some professionals) will continue to search for Fermat's 'marvellous proof' of his Last Theorem. This is despite the fact that Andrew Wiles has proved the assertion using much more sophisticated techniques than Fermat could possibly have known. (Facts drawn from Fermat's Last Theorem From Wikipedia, the free encyclopedia, and other sources.)

Question

Is there any proof in the history of mathematics that has substantially been simplified from one that only specialised mathematicians would understand to a shorter simpler proof that someone with a good understanding of maths (let's say up to beginning undergraduate level) would be able to comprehend?

In more specific terms, is there any evidence in history that supports the idea that finding a simple proof of Fermat's Last Theorem is a realistic hope?

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    $\begingroup$ I would say that MOST difficult proofs were simplified with time. In the MO (the site for professional mathematicians) just the opposite question was asked: Are there examples of, say 19s century difficult proofs that were NOT simplified. There are very few examples. $\endgroup$ – Alexandre Eremenko Sep 9 '15 at 21:51
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    $\begingroup$ It may be impossible to define the criteria for answering this question. It probably happens often that someone does a hard, long proof, and then people realize that the ideas used in the proof are of more general interest. An example of this might be the non-solvability of the quintic and Galois theory. Once you've established Galois theory, the non-solvability of the quintic might be a one-liner. $\endgroup$ – Ben Crowell Sep 10 '15 at 23:16
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    $\begingroup$ Just darwinian selection in action, easy proofs will win over complex ones. Eventually. $\endgroup$ – vonbrand Jan 16 '16 at 0:28
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    $\begingroup$ The usual course of events is that the first proof published gets simplified later by people who probably would not have found any proof themselves. $\endgroup$ – Michael Hardy Feb 26 '16 at 18:34
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    $\begingroup$ There is Gordan vs. Hilbert on invariants ... en.wikipedia.org/wiki/Paul_Gordan (regardless of whether Gordan actually said "This is not mathematics, this is theology"). $\endgroup$ – Gerald Edgar Jun 16 '17 at 0:34
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This happens all the time, mathematicians are fond of publishing papers titled Elementary Proof of Such and Such [hard] Theorem. Original proofs are often tour de force (feats of strength), they introduce new concepts along the way, make convoluted constructions, involve heavy calculations, just to get to the final result somehow. Once a new framework is developed around them ready made concepts become available, some detours are eliminated, constructions are optimized, and calculations streamlined and minimized. Sometimes a different approach allows to short circuit many original complications completely.

The prototypical example is the Pythagorean theorem, Euclid's "bride's chair" proof is a pain to follow, now there are proofs that can be understood even without words. The same goes for his volume of a pyramid proof by "method of exhaustion", even many authors writing about history prefer to replace it by a version with limits. Many Apollonius's proofs in Conic Sections are characterized (by Heath for example) as almost impenetrable even to professional geometers, but they can be streamlined using coordinate geometry. A simple proof of focal properties based on inscribing a sphere into the cone, that could have been given by Apollonius but wasn't, was only found by Dandelin in 1822. To a lesser extent the same applies to Newton's Euclidean proofs in Principia, for example his derivation of the inverse square law from Kepler's laws, a much more accessible calculus based proof is given by Bressoud in Second Year Calculus (along with an exposition of Newton's for comparison).

Abel's original proof of unsolvability of higher order equations in radicals was quite involved, Alexeev wrote a book Abel’s Theorem in Problems and Solutions based on Arnold's lectures to bright high schoolers that uses complex analysis. The same happened with the Fundamental Theorem of Algebra, elementary complex analysis proofs are now accessible to undergraduates, Gauss's original proofs are not so much. Gödel's 1930 proofs of incompleteness theorems are a technical maze (not the least because he was using notation and terminology of Russell's Principia), he simplified them himself somewhat in 1934, and now many popular books present (often somewhat sloppy) elementary proofs.

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    $\begingroup$ Very nicely broken down. Your point about accessibility to subsequent generations as the methods are simplified is well made. I get the distinct impression that each subsequent generation of humans is "smarter" than the previous generation, in part per concepts that once seemed radical or inaccessible, and are subsequently incorporated into the realm of "common knowledge", at least conceptually. (But, of course, that's just a hypothesis;) $\endgroup$ – DukeZhou Jun 15 '17 at 21:12
  • $\begingroup$ What exactly is it in Gauss' proofs that you find not accessible to undergrads? $\endgroup$ – Kostya_I Jul 11 '18 at 14:37
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The original proof of the Lasker-Noether Theorem by Emanuel Lasker took 98 pages, but now there are proofs known of less than a page. You can see it on Wikipedia.

We also have the Resolution of singularities, which was simplified from 216 pages (Hironaka, 1964) to a few dozen pages.

Also noteworthy is Ruffini's attempt at proving the Abel-Ruffini theorem, spanning over 500 pages, but it was not complete and thus cannot be considered as a proof. Abels proof was only 6 pages.

These are not really undergratuate concepts, but they are interesting (I think) for the massive decrease in number of pages.

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A proof of Fermat's Last Theorem for polynomials in one variable over the complex numbers was first found in the 19th century, using algebraic geometry. Nowadays it can be proved more simply as a consequence of the Mason-Stothers theorem, which has a very simple proof accessible to someone who knows only how to (formally) differentiate a polynomial.

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  • $\begingroup$ Care to share some links? $\endgroup$ – vonbrand Jan 16 '16 at 0:27
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    $\begingroup$ @vonbrand, I think if you google Mason Stothers Fermat you'll find some relevant links pretty soon. In any case, the treatment of the Mason-Stothers theorem in Lang's Algebra (or his Math Talks for Undergraduates) is one place to look. $\endgroup$ – KCd Jan 16 '16 at 7:26
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From Gian-Carlo Rota's book Indiscrete Thoughts (not to be confused with his other book Discrete Thoughts), pages 114–116:

One might think that once the prime number theorem was proved other attempts at proving it by altogether different techniques would be abandoned as fruitless.

[Actually, no one who knows what typically happens would think that.]

But this is not what happened after Hadamard and de la Vallée Poussin.

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for about fifty years thereafter, paper after paper began to appear in the best mathematics journals that provided nuances, simplification, alternative routes, slight generalizations, and eventually alternative proofs of the prime number theorem. For example, in the thirties, the American mathematician Norbert Wiener developed an extensive theory of Tauberian theorems that unified a great number of disparate results in classical mathematical analysis. The outstanding application of Wiener's theory, widely acclaimed throughout the mathematical world, was precisely a new proof of the prime number theorem.

[$\ldots$]

Wiener's proof had a galvanizing effect. From that time on, it was believed that the proof of the prime number theorem could be made elementary.

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It took another ten years and a few hundred research papers to remove a farrago of irrelevancies from Wiener's proof. The first elementary proof of the prime number theorem, one that "in principle" used only elementary estimates of the relative magnitudes of primes was finally obtained by the mathematicians Erdős and Selberg.

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Erdős and Selberg's proof added up to a good fifty pages of elementary but thick reasoning and was longer and harder to follow than any of the preceding ones. It did, however, have the merit of relying only upon notions that were "intrinsic" to the definition of prime number, as well as on a few other elementary facts going back to Euclid and Eratosthenes.

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It took another few hundred research papers, whittling down Erdős and Selberg's argument to the barest core, until, in the middle sixties, the American mathematician Norman Levinson (who was Norbert Wiener's research student) published a short note bearing the title "An elementary proof of the prime number theorem."

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that can be followed by careful reading by anyone with no more knowledge of mathematics than that of undergraduates at an average American college.

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    $\begingroup$ My understanding of Conway is that his laziness generally led him to "re-invent" techniques (as opposed to taking the time to look at previous work) and that this was not generally regarded as a bad thing. ;) $\endgroup$ – DukeZhou Jun 15 '17 at 21:16
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After reading other answers, I just like to add one more example. The famous prime number theorem. The proof of the theorem was given (independently) by Jacques Hadamard and Charles Jean de la Vallée-Poussin in $1896$ using ideas introduced by Bernhard Riemann, particularly, the Riemann zeta function. Now the proof was hard for someone with a undergraduate level of understanding (I am not saying that no one can!)

After almost $50$ years, in $1948$, Atle Selberg and Paul Erdős gave a rather "elementary" proof of the PNT.

See here for Selberg's paper

Wikipedia_link

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    $\begingroup$ I believe the "elementary" in Selberg-Erdös' proof is that they don't use analysis, not that it is in any sense "easy". $\endgroup$ – vonbrand Jan 29 '16 at 17:36
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    $\begingroup$ It is not that analysis is not used (quite the opposite). Their proof uses only real-variable techniques, as opposed to complex analysis. $\endgroup$ – Andrés E. Caicedo Feb 27 '16 at 5:25
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Surprising, that nobody mentions Cantor. His proof of uncountable sets of 1874 (although not as difficult as many proofs mentioned in the other answers) was not going to tempt anybody but a handful specialists. His diagonal argument of 1891 however was so easy and striking that even laymen could understand it and popularized set theory all at once.

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  • $\begingroup$ Thanks for mentioning Cantor! I find it also of interest that, like Socrates, he was accused of "corrupting the youth" because his ideas were so radical, but today the general idea of an "infinities of infinities" is popularly accepted in ideas like the multiverse. $\endgroup$ – DukeZhou Jun 15 '17 at 21:23

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