Problematic lack of intuition for number theory

Question

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?

Answer 1

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.