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I am a high school student who will be entering college in one and a half year (It is so long though).

For me, Mathematics primarily means Number Theory. My interest in Number Theory always motivates me to go and take a look at those wiki articles which are not really discussed in classic or even modern books.

For instance, while studying about prime numbers, I found many things not specifically discussed in books. Some of them are Legendre conjecture, Prime gaps, Mills constant etc. I am confused whether I should keep studying these things which authors don't dare to discuss or follow the patterns of books instead.

For your kind information, I would mention that I frequently use NZM along with G. H. Hardy's Introduction to Number Theory sometimes. For problems, I often do UC Berkeley's online assignments.

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  • $\begingroup$ Sorry, what is "NZM"? And by "Hardy's..." you mean "Hardy and Wright"? $\endgroup$
    – paul garrett
    Commented May 22, 2017 at 23:33
  • $\begingroup$ @paulgarrett NZM is for Niven Zukerman and Montgomery while Hardy's mean Hardy and wright. $\endgroup$
    – Vidyanshu Mishra
    Commented May 23, 2017 at 6:28

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Congratulations for your passion you have.

I am not a mathematician so I can't give you specific advice, but I would say go with what fuels your passion and curiosity. There is not a one-time decision, you can mix reading standard textbooks with reading some more "exotic" resources. I would suggest you combine textbooks/college courses (which are meant to give a foundation) with research papers in the field, which are usually more in depth (and also more time consuming) than textbooks. Actually, textbooks are merely a literature review over older (now considered basic) papers.

Bottom line is: dare to follow your curiosity, it can't hurt, but don't forget about the basics.

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  • $\begingroup$ Generally, I agree with these sentiments, although I fear that contemporary "research papers" in (serious) number theory often have an implicit, incompletely described, and quite daunting (if one knew...) set of prerequisites. Also, even though, yes, there is a general impulse in this direction, it is not really the case that the 100 years of number theory research prior to (e.g.) 1970 is now subsumed by standard textbooks... The main point is to not accidentally think so... (I'm waiting for some clarification as in my comments above before answering...) $\endgroup$
    – paul garrett
    Commented May 22, 2017 at 23:55
  • $\begingroup$ I don't know about the field of number theory, but in my field I noticed that there are more general papers (quite easy reading, nice overview about how to achieve a task), focused papers (they explain a concept in depth) and complex papers (put together multiple concepts and present results). The nice thing is that all of them have a lot of references which help guiding the exploration of the knowledge graph, so if the prerequisites are not met, one can go and read from references and build more knowledge before giving another shot at the paper. $\endgroup$
    – Paul92
    Commented May 23, 2017 at 0:15

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