I Have a problem. I mostly do mathematics because I find it fascinating and enjoy doing it. Now whenever I skim through a book a number theory I always find myself thinking 'I wish I would understand this because it seems so interesting'. So than I start at the beginning of the book, but I quickly loose interest.

The reason being that I just seem to lack any intuition for the subject. For things like combinatorics/analysis/graphs/linalg I all have at least a certain amount of intuition to help me in setting up proofs, and develop some sort of mental framework for the subject.

However for number theory I have absolutely no such intuition/framework. I can follow the proofs, I can prove most elementary stuff myself, but it's always very tedious. Even simple things like Bezout's Identity or Fermat's little theorem do not at all seem obvious to me.

Now I compare these results to results of a similar 'level' in other fields:

  1. The handshake lemma in graph theory
  2. A function being differentiable implies it's continuity in analysis
  3. A system of $n$ independent equalities in $n$ variables has at most $1$ solution in linear algebra

All of those seem immediately clear to me, even when I first read them. This is why I'm wondering if I'm just lacking some 'number-theory gene' or something, and I'm just not cut out to be good at number theory.

Now the question I want to ask is: to what extent is this 'normal'? Are things like Fermat's little theorem as intuitively clear to most people as for example the handshake lemma?

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    $\begingroup$ +1. I wonder why one would downvote this. I think it's a good question. As for the answer, give me some time; I'll try to type down my thoughts on this. $\endgroup$ – Kugelblitz Feb 28 '15 at 11:03
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    $\begingroup$ Do you feel similarly about modern algebra? $\endgroup$ – anon Feb 28 '15 at 17:58
  • $\begingroup$ @anon No I don't. In fact I occasionally use results from abstract algebra in proving number-theoretic stuff. However I'm always a bit uncomfortable doing that because I prefer to have intuition about this concrete case of number theory rather than having to rely on abstract and general results. In fact I often find the same result when viewed in abstract algebra quite intuitive in that framework, but when the result is translated in the specific terms of number theory it seems a lot less obvious to me. It really seems to be a case where for me abstractions make things a lot easier and clearer $\endgroup$ – user2520938 Feb 28 '15 at 19:42
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    $\begingroup$ Could you please give some examples of some number theoretic things where you seek better intuition. $\endgroup$ – Bill Dubuque Feb 28 '15 at 23:21

I think that intuition in any subject of mathematics is a skill which can only acquired only though thorough exposure to that particular subject, by familiarising oneself with common proof techniques (to that subject/field) and by working through many toy problems (exercises, if you wish). As such, nothing is "intuitively clear" to novices, and this holds for experienced mathematicians approaching an entirely new subject for the first time, too.

TL;DR: If you methodically work through number theoretical problems, by using other proofs as a guide, you should be able to acquire the understanding you seek.

  • $\begingroup$ Perusing the reference-request tag you should be able to find plenty of accessible problems. I remember seeing some notes explicitly meant to teach elementary number theory through problem solving, but I can't find them anymore. Another common suggestion is 1001 Problems in Classical Number Theory. $\endgroup$ – A.P. Feb 28 '15 at 11:28
  • $\begingroup$ Thanks for your response and the book suggestion. I'll have a look at it. It is kind of reassuring that it's not strange to not have a good intuition for a new field, not even for some of it's more elementary results. $\endgroup$ – user2520938 Feb 28 '15 at 11:54

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