What are the bottom 10 theorems of number theory? [closed]

In this article Knuth talks about the bottom 10 problems rather than top 10. The problems which are not solved but they are most ready to be solved of all.

What are they?

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closed as not a real question by Jonas Meyer, Akhil Mathew, Arturo Magidin, Qiaochu YuanDec 18 '10 at 3:06

It's difficult to tell what is being asked here. This question is ambiguous, vague, incomplete, overly broad, or rhetorical and cannot be reasonably answered in its current form. For help clarifying this question so that it can be reopened, visit the help center.If this question can be reworded to fit the rules in the help center, please edit the question.

huh? Can you clarify your question? In that article, Knuth talks about the bottom 10 questions because (a) he doesn't like this top 10 business and (b) quote "I think you've got to go for the little things, the stones that makes up the wall." I am not even sure if your interpretation (your second sentence) of what he says is what he means. And considering that the list of unsolved problems is a moving target, with the most ready to be solved ones especially so, I am convinced that this question is not a "real" question. Perhaps you should edit/rephrase/clarify. –  Willie Wong Dec 17 '10 at 20:23
@quanta: I don't thinkyour paraphrase is accurate ('problems most ready to be solved'). He says: "I don't like this 'top ten' business. It's the bottom ten that I like. I think you've got to go for the little things, the stones that make up the wall." Seems to me he is referring to problems that are at the "bottom" of the theoretical edifice, or that fill in important gaps in the theoretical edifice, the "grunt-work", rather than problems that are "most ready to be solved." –  Arturo Magidin Dec 17 '10 at 20:28
The next 10 solved problems from R.K. Guy: Unsolved problems in number theory. –  Bill Dubuque Dec 17 '10 at 20:36
@quanta: and that is not "the problems which are not solved but they are most ready to be solved of all." –  Arturo Magidin Dec 17 '10 at 21:23
@qanta: The point is that your question is very different from what Knuth is saying; you are misinterpreting his words. If you want to make that different question, then fine; the answer is simply "We don't know". If we knew what problems are "most ready to be solved" but not yet solved, we could just go ahead and solve them, thus increasing our publishing record. Knuth's point is quite different, and you seem to have missed it entirely. –  Arturo Magidin Dec 17 '10 at 23:19

Well... this is somewhat subjective but...

I think Knuth is hinting that most mathematicians (and I'm guessing most computer scientists) work on problems that are not particularly glamorous and not immediately important, nor do they have particularly elegant proofs.

The use of the phrase "bottom ten" appears to be a metaphor, referring to problems that could be solved (using well-established techniques), but haven't yet been solved. There's no actual list.

The "stones that make up the wall" refers to that important mathematical discoveries are typically not made by one person working hard on a problem, but by a host of contributors over many years, each making incremental progress on the problem.

If I have seen a little further it is by standing on the shoulders of Giants. -- Isaac Newton

PS. I can understand that Knuth's response might be surprising to people without research experience. Why would he want to work on the least interesting problems?

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