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Stillwell mentions in his book Mathematics and its History that:

Most of the really old unsolved problems in mathematics, in fact, are simple questions about the natural numbers...

What is it about Euclidean geometry that makes it have so few classical unsolved problems, relative to number theory?

*** Inspired by this question, Why are so many of the oldest unsolved problems in mathematics about number theory? and motivated by this comment.

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bzm3r
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    Geometry is decidable, number theory is not. – Asaf Karagila Feb 03 '14 at 03:01
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    @AsafKaragila Could you explain what you mean by "decidable" in a way that could make sense to me? – bzm3r Feb 03 '14 at 03:02
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    @Asaf: That was my first thought as well, but isn't the fastest known algorithm something like doubly exponential (in the length of the statement) in run time? That makes it seem like practically speaking, the fact that geometry is decidable is not so relevant. But I'm way outside my comofort area here... – Jason DeVito - on hiatus Feb 03 '14 at 03:12
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    @Jason: Yes, and it is in fact not directly relevant. But I think that decidable theories are inherently better understood than undecidable theories. Because it means exactly that a computer could easily (even if it would take a very long time) identify all the theorems from the non-theorems. twirlobite: The modern axiomatization of geometry is such that a computer software could take as an input a statement in the language of geometry and tell us if it is a theorem or not. The algorithm is very very slow, but it exists nonetheless. – Asaf Karagila Feb 03 '14 at 03:18
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    (I should add that this is just a gist, and I really don't know enough about either theory except for the basics things on them. But I do think that decidability makes a difference in how complicated the theory is for us to understand. Maybe I'm wrong... that's why it's just a comment, and not an answer.) – Asaf Karagila Feb 03 '14 at 03:20
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    You picked a time to ask that when Euclidean geometry was out of long-standing open problems. The classical construction problems and the problem of the fifth postulate lasted about two thousand years. Do you think the Riemann hypothesis or P = NP will hold up that well? – bof Feb 03 '14 at 03:50
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    Perhaps the difference is that today's geometry is something Euclid (and perhaps even Gauß) wouldn't recognize as such (in terms of problems posed, not solutions)? – vonbrand Feb 03 '14 at 04:20
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    @bof: If one takes a Platonic point of view, then P=NP and Riemann Hypothesis both existed, as questions, at least as long as Euclid's fifth postulate problem... :-) – Asaf Karagila Feb 03 '14 at 08:04
  • @JasonDeVito No, by Godel's theorem and subsequent work any deductively complete formal system like EG (Euclidian Geomtery) has a lexical truth function that given any syntactically valid statement in the formal system, transforms it into an arithmetical expression that always returns 0/1 based on the truth of the statement. Both the transform and the expression are closed and so run in O(n) steps where 'n' is the length of the statement. I believe that EG's function was demonstrated back in the early 60's. As such, EG will never again have an unsolved problem, and is thus "uninteresting". – RBarryYoung Feb 04 '14 at 14:52
  • @vonbrand Yes, unlike formalized Euclidian Geometry modern "interesting" branches of geometry, such as Projective Geometry, etc. extend (and/or modify) the Euclidian concepts a great deal and are not deductively complete and thus still fall into the realm of exponential proof searches. – RBarryYoung Feb 04 '14 at 15:02
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    @RBarryYoung: I somehow feel that your two last comments, with a bit of fleshing out and a few more references, would be an awesome answer to this question. – Willie Wong Feb 04 '14 at 15:25
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    @WillieWong Yeah, but its been 35 years since I worked on any of this. I am having a lot of trouble finding the references, or even recalling the proper mathematical terms to search for. :-( I'll try though. – RBarryYoung Feb 04 '14 at 15:30
  • @RBarryYoung Perhaps this will help? http://math.stackexchange.com/questions/90393/why-euclidean-geometry-cannot-be-proved-incomplete-by-godels-incompleteness-the – bzm3r Feb 04 '14 at 19:44
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    @AsafKaragila I would appreciate a lot if someone could point to resources that explain why Euclidean geometry is decidable and/or projective geometry is not. – Phira Feb 04 '14 at 23:23
  • @RBarryYoung I guess that you and Asaf are the most likely to help me on this. – Phira Feb 04 '14 at 23:24
  • @Phira: I don't really know the details of these results, just the results themselves. I suppose that a Ouija board might come in handy if you can channel Alfred Tarski to explain that to you. :-) – Asaf Karagila Feb 04 '14 at 23:28
  • After some research I am no longer confident that my earlier statement about being resolvable in O(n) is correct. I had forgotten a few things (such as recursive enumeration, and that linear optimization problems can be restated as a problem of planar inequality bounding maximums), and now believe that I may have misunderstood the potential complexity of the transformation procedure. – RBarryYoung Feb 05 '14 at 15:05
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    @Phira: This is discussed in Old and New Results in the Foundations of Elementary Plane Euclidean and Non-Euclidean Geometries, see sections 3.2 and 3.3. Compared with other axiom sets for Euclidean geometry, Tarski's is clever but crippled. It can't even refer to sets of points. Other axiom sets yield undecidable theories even for elementary Euclidean plane geometry. Just because the comment "Geometry is decidable, number theory is not" got a bunch of upvotes doesn't make it correct. It is not correct. – Matt May 14 '20 at 21:07

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Because Euclidean geometry is currently not fashionable, most people do not study topics in it or discuss problems in it, and so you simply hear of fewer problems, solved or unsolved.

Any claims that all Euclidean geometry problems are decidable, as given in the comments to the question, will depend on some restricted definition regarding the form that a "problem" can take.

There are plenty of unsolved geometry problems. I recommend the book Unsolved Problems in Geometry by Croft, Falconer, & Guy (1991). In addition to hundreds of problems, the book even points to 17 other collections of unsolved geometry problems.

Many of these unsolved problems are fairly easy to state. For example: Among all configurations of $n$ points not all on one line, what is the minimum number of lines they might determine? Explainable to a small child, yet unsolved.

There are many, many beautiful unsolved geometry problems, including ones with Erdös rewards, and I cannot do them justice in this answer.

Matt
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    Are you sure that you fully understand what the term "decidable" means? It's a technical term, rather than the natural meaning of the word which one can understand as "can be solved". – Asaf Karagila Feb 26 '14 at 20:44
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    @AsafKaragila Yes, I am sure, I have published papers on the topic. – Matt Feb 26 '14 at 20:53
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    That's generally hard to tell from just "Matt". So excuse me for not knowing that. :-) – Asaf Karagila Feb 26 '14 at 21:01
  • No problem! :-) – Matt Feb 26 '14 at 21:02
  • @AsafKaragila To add a bit of content, you could imagine someone describing a Turing machine (or a proof formalism) in geometrical terms (like how Gödel numbering lets you describe such things in terms of arithmetic), so that a "geometry" problem is equivalent to the halting problem, or to a statement that the statement has no proof, etc. Of course this geometry problem will not be of the limited form which has been classified as decidable. – Matt Feb 26 '14 at 21:17