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What are some easily-stated relatively recently proven theorems? I don't mean they were necessarily easy to prove, just easy to state. Here are a few examples:

  • The proof of Fermat's Last Theorem was completed in 1995, according to Wikipedia, by Wiles and others.

  • Green and Tao proved that for any $N$, there is an arithmetic sequence of primes of length at least $N$.

  • This year, Yitang Zhang made progress toward the twin prime conjecture - he found an integer $K$ such that there are infinitely many pairs of distinct primes that differ by less than $K$. I think that $K$ was in the neighborhood of 17 million or so but lower bounds were found within months. Sorry I don't have more specifics; see Yitang Zhang: Prime Gaps.

  • According to http://truthiscool.com/prime-numbers-the-271-year-old-puzzle-resolved, Helfgott has proved the weak Goldbach conjecture (any odd integer >5 is the sum of 3 primes (that is the wording from the article, I apologize if it is imprecise or wrong). The article states "Helfgott's preprint is endorsed and believed to be true by top mathematicians, Tao among them". The article is old (May 13, 2013) and I don't know if the result has been peer-reviewed and published in a journal. The conjecture is easy to state and if the proof is indeed valid it belongs on the list.

Notice that all four theorems above are in number theory (the statements of the theorems, anyway. The proofs may have used stuff from other branches of mathematics, I don't know.)

  • Fairly recently, some young folks found a deterministic primality-testing algorithm that had polynomial computational complexity (in time). One might consider this more of a theoretical computer science result than a mathematics result. Again, sorry, I forgot the specifics.

  • I think Perelman's proof of the Poincare Conjecture almost qualifies. It is difficult to explain exactly what it means for a manifold to be orientable, even to most mathematicians, let alone laymen.

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Stefan Smith
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    That result of Tao is actually a result of Green and Tao. – Elchanan Solomon Sep 27 '13 at 02:24
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    It was Yitang Zhang who made the breakthrough on the twin primes problem. This sparked Tao's twin primes polymath project which improved on Zhang's result. – littleO Sep 27 '13 at 03:00
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    The *weak* Goldbach conjecture was apparently proved by H. Helfgott this year. – Jeppe Stig Nielsen Sep 27 '13 at 06:31
  • @IsaacSolomon : thanks, I corrected my omission. – Stefan Smith Sep 27 '13 at 13:40
  • @littleO : thanks, I edited the question. – Stefan Smith Sep 27 '13 at 13:41
  • @JeppeStigNielsen : this would have made a nice answer rather than a comment. I added it to the list in my question. – Stefan Smith Sep 27 '13 at 13:55
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    Dear Stefan, There is no reason to doubt Helfgott's result, but note that this result builds on the fundamental contributions of Vinagrodav from over 70 years ago; the difference with Helfgott's result is that Vinagradov proved ternary Goldbach for all odd numbers greater than an unspecified bound $N$, while Helfgott's work replaces $N$ by $5$. Regards, – Matt E Sep 27 '13 at 14:02
  • @MattE : Thanks for the info. I'm doing actual work now, but when I have time, I'll investigate Vinagradov's work. – Stefan Smith Sep 27 '13 at 14:09
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    @StefanSmith: Dear Stefan, You might also be interested in Terry Tao's post where he explains his proof of a weaker result (sum of at most five primes, rather than three primes) but again, for all odd numbers. I think Helfgott's argument was inspired by Tao's. Regards, – Matt E Sep 27 '13 at 16:37
  • Wiles's proof is a significant accomplishment, but can he fit it into the margin of Diophantus' Arithmetica? :) Also, I seriously doubt that this is the theorem Fermat claimed to have. I'm still waiting for a theorem that could lay credible claim to being Fermat's solution to the problem. – MitchC Sep 03 '15 at 01:18
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    The consensus seems to be that Fermat didn't have even a proposed proof of FLT, just some special-case evidence. I think that even the best-case scenario is that he had some form of Kummer's factorization-based proof but didn't realize that there could be irregular primes. – Steven Stadnicki Sep 03 '15 at 01:22
  • I hope that most mathematicians know what an orientable manifold is, or at least would easily understand an explanation of the concept. – Taladris Dec 09 '17 at 11:15

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Catalan‘s conjecture (a.k.a. Mihăilescu’s theorem):
http://en.wikipedia.org/wiki/Catalan%27s_conjecture

Kieren MacMillan
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Well, I'm not sure if this has been confirmed yet, but apparently in March, Ciprian Manolescu claims to have refuted the Triangulation Conjecture in dimensions $ \geq 5$. It's not the simplest result to state, but it's not terribly technical (unlike the proof, I imagine). The conjecture essentially states that "every compact topological manifold can be triangulated by a locally finite simplicial complex," in the language of the linked article. Put less rigorously, you can't necessarily take a nice, compact manifold of high dimension and cut it into triangles that fit together like puzzle pieces.

Elchanan Solomon
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    Thanks. I upvoted your answer. I looked at the link. The write-up was amusing and the result w as interesting. But any result with the word "homology" in the title can't fit on my list unless you are being very generous. – Stefan Smith Sep 28 '13 at 16:45
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There are infinitely many pairs of primes $p_1 < p_2$ such that $$p_2 - p_1 < \text{(some specific large number)}.$$

Daniel McLaury
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