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I have completed studying Galois theory, Fermat's Last Theorem for Regular prime and some number theoretic complex analysis (prime number theorem), and basic linear forms in logarithm.

What else should I complete reading to be eligible to read contemporary literature in Diophantine Equation (Exponential) ?

My graduation was in engineering, so I am from non-math (I mean serious math!) background, and I am not going to a university in near future, but wish to conduct research as an independent researcher.

There might be a lack of specification in my question, so if possible, adjust your answer according to that, also feel free to edit.

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    $\begingroup$ Try looking at papers at a number theory journal, such as Journal of Number Theory, Acta Arithmetica, International Journal of Number Theory, and Algebra and Number Theory and see if you can understand them. If not, then check their references and read those, continue until you reach foundational texts written in a language that you can understand $\endgroup$
    – Stanley Yao Xiao
    Commented Aug 8, 2020 at 18:07
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    $\begingroup$ The modern study is based a lot upon algebraic geometry. So the more of this you know the better. $\endgroup$
    – Daniel Loughran
    Commented Aug 8, 2020 at 19:00
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    $\begingroup$ This book (Rational Points on Varieties by the irrepressible Bjorn Poonen) contains a lot of the prerequisites you would need in case you go the algebro-geometric route: www-math.mit.edu/~poonen/papers/Qpoints.pdf $\endgroup$
    – R.P.
    Commented Aug 8, 2020 at 21:01
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    $\begingroup$ However, the learning curve for the geometric track is both steep and long (a true via longissima). The "pay-off" of your studies (meaning concrete results and papers of your own) would probably be greater if you wouldn't concentrate exclusively on all that high-falutin stuff, and set your sights more on analytic number theory, additive combinatorics, recreational problem solving... There is always a severe danger of involving yourself in massive propaedeutic studies, without the slightest guarantee of the possibility of contributing something of your own. Just my two cents... $\endgroup$
    – R.P.
    Commented Aug 8, 2020 at 21:09
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    $\begingroup$ I don't think there is a coherent field of mathematics called Diophantine equations. It's like classification of finite simple groups: a statement about objects that are very easy to define but its proof consists of numerous subcases involving absolutely disparate techniques that have nothing to do with each other. I expect that the vast majority of diophantine equations can not be meaningfuly attacked with the techniques of algebraic geometry, Galois representations and automorphic forms. So there are no well-defined prerequisites. $\endgroup$
    – user158636
    Commented Aug 9, 2020 at 11:20

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If I understand correctly, the keywords for the type of number theory you're interested in are "Diophantine approximation" and "transcendental number theory." Are you aware of Florian Luca's article on Exponential Diophantine Equations? If not, that might be a good place to start. You might also like M. Ram Murty and Purusottam Rath's book Transcendental Numbers.

Getting to the research frontier is always a tricky business. Michel Waldschmidt's Diophantine Approximation on Linear Algebraic Groups will give you a sense of what some of the major open problems in the subject are, but that's a very demanding book. There undoubtedly exist lower-hanging fruit, but it may be hard to find such fruit without an experienced advisor to guide you.

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  • $\begingroup$ maybe it is useful to remark that even ignoring the issue of an advisor, one valuable source of information that a "normal" graduate student has access to and an independent researcher doesn't is a group of peers at roughly the same level of mathematical understanding as you that you see on a day-to-day basis. I think for many people the perspective of other people allows them to re-gauge their views on what is important/difficult and to lose any misconceptions sooner than they might otherwise. $\endgroup$
    – user158636
    Commented Aug 10, 2020 at 6:21
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    $\begingroup$ +1 But I don't know how I feel about referring an independent (that is unpaid) researcher to an article behind a paywall, sold by price-gouging publishing houses. I would try to e-mail the author and request a preprint copy from him, if I were the OP. $\endgroup$
    – R.P.
    Commented Aug 10, 2020 at 9:36
  • $\begingroup$ @RP_ : Is it behind a paywall? I thought it wasn't, because I am able to view it from a machine that isn't registered to my institution, but maybe I have some cookies in my browser or something. $\endgroup$
    – Timothy Chow
    Commented Aug 10, 2020 at 13:25
  • $\begingroup$ @TimothyChow I think that must be the case. The page I'm getting says I can buy the chapter for 27,20 EUR (which isn't that bad comparatively speaking, but still). $\endgroup$
    – R.P.
    Commented Aug 10, 2020 at 13:36
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    $\begingroup$ Is Google Books any better? $\endgroup$
    – Timothy Chow
    Commented Aug 10, 2020 at 14:03

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