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At the moment, are there any major unsolved mathematical problems yet to be solved, and do they have any prize associated with the solving of them?

Furthermore, is there any particular reason that they have not yet been solved?

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closed as too broad by Thomas Andrews, user296602, Asaf Karagila, Chris Godsil, Will Jagy May 8 '16 at 18:51

Please edit the question to limit it to a specific problem with enough detail to identify an adequate answer. Avoid asking multiple distinct questions at once. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.

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    $\begingroup$ This question is way too broad. What defines "hardest?" Asking for explication is also a problem, because a rathole of detail can be gone down, while still not being useful. Why is the Collatz conjecture hard? Some will give the simple argument that it is hard because the interplay between addition and multiplication is tricky. But it might hard because the smallest counter-example has $10^{100}$ digits, or it might be hard because it is unresolvable in Peano arithmetic. $\endgroup$ – Thomas Andrews May 8 '16 at 17:24
  • $\begingroup$ I agree with @ThomasAndrews that there is no one, particular, reason they are all hard. Each is unique. I have a vague-ish historical explanation in my post, but it may not be what you are after. $\endgroup$ – user237392 May 8 '16 at 17:41
  • $\begingroup$ en.wikipedia.org/wiki/List_of_unsolved_problems_in_mathematics $\endgroup$ – Rahul May 8 '16 at 17:42
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    $\begingroup$ Every unsolved problem is potentially the hardest problem. Until you solve all the problems, you won't know which problems were easier and which were harder. What about problems which are easy to solve, granted you solved something very difficult? Are they considered very easy, or harder than the difficult problem (since they required more effort)? This question is entirely subjective and way too broad. $\endgroup$ – Asaf Karagila May 8 '16 at 17:42
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Any answer to your question will risk overgeneralizing. However, taking a look at the great problems of antiquity, one theme that occurs is the that we lacked the proper language to express what a solution would even look like.

For example, the Geometric Problems of Antiquity were insoluble given that they were expressed using the language of straightedge and compass. Once we moved to the more abstract algebraic approach, these ceased to be issues, and became, in a sense, trivial.

Another set of problems involved problems of infinite processes before calculus and real analysis. These are best described by Zeno's Paradoxes.

A quick glance at the Clay Millenium Problems shows they are a diverse set, so I doubt there is any one "thing" making them hard. However, again at the risk of overgeneralizing, they are unsolved because we cannot express the properties of a possible solution (note: not "the" solution, but the criteria that make it a possible solution.)

As a concrete example. In differential equations, we know the solution must be a function of the given arguments. A more abstract one is Wiles' proof of Fermat's Last Theorem: he re-cast it as a problem in the theory of modular elliptic curves, whose solutions require certain conditions, and, with a TON of ingenuity and intelligence thrown in (of which I cannot ever hope to comprehend) he shows that Fermat's Last Theorem is true. From what I can understand of his approach, it was the great insight the solution would be within the framework of elliptic curves.

I really can't make this more specific given your question and my limited experience with specific unsolved problems. My answer comes from my reading of various histories of mathematics, where previously "unsolvable" problems have now become standard fare.

However, given the number of very smart people working on them (unsuccessfully up to this point), it appears that the solutions are not "first order", where we know the outlines of what we need to do but lack a specific solution; instead, they are "second order" where we don't even know what a solution would imply or what properties it would have.

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The Millennium Prize Problems | Clay Mathematics Institute $1 million allocated to each

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    $\begingroup$ That makes them important, but not any harder than other open problems. $\endgroup$ – Asaf Karagila May 8 '16 at 17:43
  • $\begingroup$ They are harder. These are not randomly selected, but selected from Mathematician David Hilbert list of twenty-three problems for their importance to mathematics. The best mathematicians in the world have tried in vain for centuries to prove/disprove them, so definitely that makes them harder. $\endgroup$ – Simple May 9 '16 at 1:57
  • $\begingroup$ That is a terrible appeal to authority. The fact that there are problems that the best mathematicians of the last 120 years didn't tackle, doesn't mean that these problems are not even harder than any of the Millennium ones. $\endgroup$ – Asaf Karagila May 9 '16 at 13:03

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