# Problematic lack of intuition for number theory

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?

• +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. – Kugelblitz Feb 28 '15 at 11:03
• Do you feel similarly about modern algebra? – anon Feb 28 '15 at 17:58
• @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 – user2520938 Feb 28 '15 at 19:42
• Could you please give some examples of some number theoretic things where you seek better intuition. – Bill Dubuque Feb 28 '15 at 23:21