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NOTE: I will not accept an answer because a lot of answers are really good and if anyone want to post under this question later then they are most welcome to post as comment or answer because it will certainly help many people. Thanks to all that contributed.

I have completed my master's in mathematics a couple of years ago and due to very strong personal and professional reasons I couldn't get admitted to grad school despite having a good academic background.

I have a really good background in number theory. I don't have guidance of any professor right now and I want to try working on an open problem.

Can you please let me know of resources( websites/ blogs/books) where I can find open problems in number theory to work on along with their estimated level of difficulty ( if possible)?

Thanks!

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    $\begingroup$ For my various open conjectures in number theory, you may visit my homepage maths.nju.edu.cn/~zwsun . $\endgroup$
    – Zhi-Wei Sun
    Commented Apr 26, 2022 at 15:49
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    $\begingroup$ might it be possible that open conjectures exist which are not "difficult" to prove? a more productive strategy I would imagine is to identify a research direction where the open conjectures have not yet been formulated, so you can hope to find a conjecture that is both interesting and not too difficult; it is likely that you will need the guidance of an experienced researcher to find such a promising research direction, it's not something you will get from the web. $\endgroup$
    – Carlo Beenakker
    Commented Apr 26, 2022 at 16:21
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    $\begingroup$ If a problem is open, then how is anyone to know its level of difficulty? $\endgroup$
    – Steven Landsburg
    Commented Apr 26, 2022 at 23:45
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    $\begingroup$ Any problem that is clearly formulated, clearly publicised, and still open, is almost inevitably going to be difficult. (Proof: Any problem that’s clear, publicised, and not difficult will swiftly get solved.) $\endgroup$
    – Peter LeFanu Lumsdaine
    Commented Apr 27, 2022 at 8:26
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    $\begingroup$ You could search this very website (and also math.stackexchange) for unanswered questions tagged number-theory. You can also have a look at the problem sets from the annual West Coast Number Theory meetings, posted at westcoastnumbertheory.org/problem-sets $\endgroup$
    – Gerry Myerson
    Commented Apr 27, 2022 at 13:46

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There is the famous Unsolved Problems in Number Theory by Richard Guy, 3rd Edition, 2004. PDF available at the Springer link.

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    $\begingroup$ Springer should perhaps try to convince several number theorists to update the book to a 4th edition. $\endgroup$
    – Joseph O'Rourke
    Commented Apr 27, 2022 at 16:27
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    $\begingroup$ @TimothyChow For example, the discussion around the Goldbach conjecture has been affected by Helfgott's work on ternary Goldbach. Looking through a couple chapters, I would concur that most likely a majority have not seen progress. $\endgroup$
    – Will Sawin
    Commented Apr 27, 2022 at 17:42
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    $\begingroup$ Noga Alon tells an interesting story about Guy's book (see page 7). The first time he picked up the book, he opened it randomly and read a problem that he had never seen before. He solved it fairly rapidly. Encouraged by this success, he combed through the book, but was unable to make any progress on any other problem. $\endgroup$
    – Timothy Chow
    Commented Apr 27, 2022 at 20:39
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    $\begingroup$ @Timothy, problem E19 asks, "Are the integer parts of the powers of a fraction infinitely often prime?" Guy notes that Forman and Shapiro proved that $[(3/2)^n]$ and $[(4/3)^n]$ are composite infinitely often. Since then, Dubickas and others have found more examples on the composite side of the question. See my answer at mathoverflow.net/questions/153426/… $\endgroup$
    – Gerry Myerson
    Commented Apr 28, 2022 at 1:02
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    $\begingroup$ Problem A8 asks about gaps between primes, in particular, about twin primes. In 2013, Yitang Zhang produced a tremendous improvement on previous results, showing there were infinitely many pairs of primes separated by at most $70,000,000$. That number has been reduced to $246$. The 1st edition of Guy's book asked, "are there six points in the plane, no three on a line, no four on a circle, all of whose mutual distances are rational?" Such a set having been found by Leech, the 3rd edition asked "are there any sets of more than six such points?" (continued) $\endgroup$
    – Gerry Myerson
    Commented Apr 28, 2022 at 1:22
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To some extent, research questions are really research themes, and these can definitely be found in journals. Just open a journal (e.g. Algebra and Number Theory) and read an article or review. Sometimes picked at random it can be interesting to read to the end, and you're left with many ideas for new developments, and often direct open questions stated at the end of the paper.

Attacking the "open problems" (as they are called) in number theory (e.g. Riemann etc) straight away may not be fruitful as they can be very difficult if attacked directly from first principles, though this is tenacious and can teach you a lot about the problem. Though to some extent almost no one is just attacking them. Even Wiles was working on something else when he realized he could therein work to prove Fermat's final theorem as a consequence of what he'd seen. He was working on the arithmetic of elliptic curves with complex multiplication by the methods of Iwasawa theory, which had surrounding workers publishing about it (from what I understand).

But there are plenty of research themes in number theory (and otherwise) which are currently being worked on (e.g. probabilistic number theory, etc), and I suppose one would look at research articles in journals of number theory to see what those are. What are other students in number theory working on? What did they publish? Even take a recent PhD thesis of interest and ask "what can be done to develop these results", or "why are these results not interesting"?

Knowing all the research themes is quite difficult, but one learns about them in conferences etc, sometimes presented by experts, but more often presented by PhD students just learning the ropes of the area, each are useful to listen to. Each is a window into a research area one could pick up and get into, like an investment or stock, which hopefully pays off in the long run.

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Go to a good math library, pick up a few books (perhaps even randomly) in number theory and start reading them until you find one book which you really like. I am (almost) sure that you will ask yourself plenty of questions when studying deeply some interesting mathematics. At the beginning, many of these questions are already solved (and perhaps not hard) but questions help advancing. At some point, some of your questions will have no known answer. With some luck, one of these questions is interesting and solvable. (Books can of course be replaced by attendance of lectures, discussions with collegues etc.)

The point is that one needs often some external input and good books can provide this.

A last piece of advice: Do not chose the crowdest corners (e.g. around Riemann hypothesis), at least at the beginning, if you want to make some contributions.

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Do check out the Handbook of Number Theory (Volumes I and II), if you need to refer to a compendium of the latest results on number theory. (I forget which Volume has open problems.)

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