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Vanadium 50

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If they were easy, they'd be solved.but I want to find some unsolved problems on these fields but not very difficult

- #4

phyzguy

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https://projecteuler.net/

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There is a million dollars on offer if you can solve any of these:

https://en.wikipedia.org/wiki/Millennium_Prize_Problems

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etotheipi

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$$P = NP$$ $$\frac{1}{P} \times P = \frac{1}{P} \times NP \implies N=1$$ Where's my million dollars? I've gotta admit I'm slightly surprised no one has solved that before.There is a million dollars on offer if you can solve any of these:

https://en.wikipedia.org/wiki/Millennium_Prize_Problems

- #8

member 587159

What if ##P=0##?$$P = NP$$ $$\frac{1}{P} \times P = \frac{1}{P} \times NP \implies N=1$$ Where's my million dollars? I've gotta admit I'm slightly surprised no one has solved that before.

- #9

etotheipi

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Blast, now I'm going to need to retract my email to the Clay Institute.What if ##P=0##?

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phyzguy

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phyzguy

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And you want FIFTY? If you can solve one, it will make your career. The Millenium Prize Problems in Post #6 is a good place to start. Why don't you start with the Riemann Hypothesis? It's been around for over 150 years and is still unsolved.To phyzguy. I mean problems that nobody knows how to solve.Where can I find some of them?

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If the problems are too hard, hard enough to be the subject of a PhD thesis, then I guess there is no generic process on solving them, other thanWhat are the general or specific processes to solve such problems?

- Understanding very well the theory (know and understand all the theorems and definitions that are related) behind the problem
- Understanding what the problem asks and what the problem gives
- Trial and error as you say of various attacks to the problem. The solution to the problem might be too hard and long (for example I think that the proof to Fermat's Last Theorem is over 100 pages). In this case there is nothing you can do other than study hard, invent various methods and attacks to the problem guided by intuition or by experience from other known solutions to similar problems. Inspiration might be the key also, if the problem is too hard. There are problems that their solution is not too long but they require some sort of specific trick or some "key" that unlocks the solution, and to find that key either you have to rely on inspiration or in experience .

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I would add to your list:If the problems are too hard, hard enough to be the subject of a PhD thesis, then I guess there is no generic process on solving them, other than

- Understanding very well the theory (know and understand all the theorems and definitions that are related) behind the problem
- Understanding what the problem asks and what the problem gives
- Trial and error as you say of various attacks to the problem. The solution to the problem might be too hard and long (for example I think that the proof to Fermat's Last Theorem is over 100 pages). In this case there is nothing you can do other than study hard, invent various methods and attacks to the problem guided by intuition or by experience from other known solutions to similar problems. Inspiration might be the key also, if the problem is too hard. There are problems that their solution isI not too long but they require some sort of specific trick or some "key" that unlocks the solution, and to find that key either you have to rely on inspiration or in experience .

4. Spontaneous inspiration, like silently staring at a problem until the answer comes suddenly out of the blue (Feynman's technique) or seeing the solution in a dream.

Unfortunately, these techniques are not very reliable.

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1. AI will we have AI soon will computer be able to make decisions and replace humans at job.This is kind of related to machine learning. Some people have done AI to make computer be able to play tetris very well. Also this is releated to moore law can we make the transistors and electrical components smaller.

2. How long can human live can we develop technology that can make humans live much longer and reverse the aging process. I think this will get a lot of interest in the future.

3. Artificial womb can humans be grown in pods filled with nutrients kind of like in the novel brave new world.

4. renewable energy sources can solar power be made more efficient.

- #16

mathwonk

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As others have said, any problem that is publicly well known and interesting, is likely to be either quickly solved by interested and knowledgable parties, or to be very difficult, having resisted efforts of such people. Thus the ticket to finding and solving problems of reasonable difficulty, is to learn to generate them oneself. For this, you want to learn as much as possible, and practice asking questions that come to mind in thinking about this new knowledge. You can start this exercise in reading standard textbooks, but once you progress beyond those to more advanced works, it will help to choose to read the work of very strong and creative people. Such works will contain not only very useful insights, but also many newly uncovered areas and questions. It is true that sometimes such a strong and creative researcher will keep somewhat quiet about some of his/her discoveries in order to give them to his/her own students, to give them a head start, but it is best if a student develops his/her own approach. In this way the problem will actually be interesting to the student.

It is also true that one should not be too concerned as to whether the problem one generates has already been solved or not. it is just as impressive to discover and solve a non trivial problem on ones own, even if it has already been solved, and indeed there is always room for a new approach even to an already solved problem. Also, if one persists in this endeavor, at some point one will usually happen on a problem that is new. When I was a grad student, I solved two or three problems that turned out to have been already solved, before finally making progress on one that was not solved. Such problems are sometimes found in the publications of good mathematicians from the past. (My thesis problem was an open question revealed on virtually the last page of a paper by Wirtinger from the 19th century.) So my advice is again, practice generating questions from all ones reading, as well as making ones own proofs of results one is learning, and definitely make it a habit to read very good mathematicians. Even the best mathematicians spent over a century essentially working out remarks of Riemann.

Another technique for solving new problems is to understand thoroughly the proof of a known theorem, and to observe that it does not use all the hypotheses that are present. Hence one can prove a similar theorem by omitting that hypothesis and using the same argument. Or, as my teacher Maurice Auslander told us, "look for a theorem whose proof proves more than it claims to". Or as Zariski himself did, and this is more difficult, read a paper but not the proofs, and try to give your own proofs of the results. Then you will sometime find that your proof proves more than the statements you are reading. Another technique, after acquiring some level of mastery: one may read old papers by good people of an earlier era, whose results are interesting, but sometimes lacking in modern clarity or rigor, and one can sometimes use better techniques than were previously known, to clarify and solidify, and possibly improve, their results.

Basically we try to build upon the work of others, and to choose those others wisely, so as to be following in the footsteps of someone as good as possible, but within reason, for our own ability. As we do this moreover, our ability increases.

Another good possibility is to work jointly with other people. Most of my work, and all of my best work, is joint with some very strong people, and the results are much better than anything I did alone. People with different strengths can compliment each other very profitably. You also learn a lot by this process of collaboration.

It is also true that one should not be too concerned as to whether the problem one generates has already been solved or not. it is just as impressive to discover and solve a non trivial problem on ones own, even if it has already been solved, and indeed there is always room for a new approach even to an already solved problem. Also, if one persists in this endeavor, at some point one will usually happen on a problem that is new. When I was a grad student, I solved two or three problems that turned out to have been already solved, before finally making progress on one that was not solved. Such problems are sometimes found in the publications of good mathematicians from the past. (My thesis problem was an open question revealed on virtually the last page of a paper by Wirtinger from the 19th century.) So my advice is again, practice generating questions from all ones reading, as well as making ones own proofs of results one is learning, and definitely make it a habit to read very good mathematicians. Even the best mathematicians spent over a century essentially working out remarks of Riemann.

Another technique for solving new problems is to understand thoroughly the proof of a known theorem, and to observe that it does not use all the hypotheses that are present. Hence one can prove a similar theorem by omitting that hypothesis and using the same argument. Or, as my teacher Maurice Auslander told us, "look for a theorem whose proof proves more than it claims to". Or as Zariski himself did, and this is more difficult, read a paper but not the proofs, and try to give your own proofs of the results. Then you will sometime find that your proof proves more than the statements you are reading. Another technique, after acquiring some level of mastery: one may read old papers by good people of an earlier era, whose results are interesting, but sometimes lacking in modern clarity or rigor, and one can sometimes use better techniques than were previously known, to clarify and solidify, and possibly improve, their results.

Basically we try to build upon the work of others, and to choose those others wisely, so as to be following in the footsteps of someone as good as possible, but within reason, for our own ability. As we do this moreover, our ability increases.

Another good possibility is to work jointly with other people. Most of my work, and all of my best work, is joint with some very strong people, and the results are much better than anything I did alone. People with different strengths can compliment each other very profitably. You also learn a lot by this process of collaboration.

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@Math_QED:$$P = NP$$ $$\frac{1}{P} \times P = \frac{1}{P} \times NP \implies N=1$$ Where's my million dollars? I've gotta admit I'm slightly surprised no one has solved that before.

"i will end this man's whole career"

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- #19

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Well, if ##P=0##, then ##N## can be anything. I think we can start writing the paper together. I'll be second author since you did most of the work :)I mean I spent a solid 30 to 40 minutes formulating that proof, so it's fair to say that @Math_QED ruined my afternoon.

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Keith_McClary

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Usually called open problem.

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- #22

Mark44

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If you are an undergraduate who has trouble passing class exams in mathematics or physics, as I believe you have stated in other threads, the chances of you solving open problems in either of these fields is effectively zero.

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You're not even posting reasonably correctly. For example, you don't put a space after a sentence-ending punctuation mark. Also, you don't bother to capitalize the letter 'i' when referring to yourself.

There is not a basket of unsolved problems each of which could be immediately solved by a person who needs only to be told of where the basket is located.

Once in a great while, some brilliant person solves a previously unsolved problem; however, the unsolved problems are unsolved because they are either unsolvable or exceedingly difficult ##-## in some cases, we're not sure which ##\dots##

- #24

Vanadium 50

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If you want to solve problems so hard nobody has solved them yet, you really want to be smart.

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