Reddit mentions of A Course in Arithmetic (Graduate Texts in Mathematics, Vol. 7)

I recommend this as the new universal elementary school math book.

Sometimes they literally don't know any mathematics, so I teach them some arithmetic.

That's too dificult. Start with some arithmetic.

Well, Hardy & Wright is the classic book for elementary stuff. It has almost everything there is to know. There is also a nice book by Melvyn Nathanson called Elementary Methods in Number Theory which I really like and would probably be my first recommendation. Beyond that, you need to decide which flavour you like. Algebraic and analytic are the big branches.

For algebraic number theory you'll need a solid grounding in commutative algebra and Galois theory - say at the level of Dummit and Foote. Lang's book is pretty classic, but maybe a tough first read. I might try Number Fields by Marcus.

For analytic number theory, I think Davenport is the best option, although Montgomery and Vaughan is also popular.

Finally, Serre (who is often deemed the best math author ever) has the classic Course in Arithmetic which contains a bit of everything.

A graduate student in algebra should know Galois theory intimately, or else the course is Galois theory. Galois theory is akin to required calculus in undergraduate, especially for pure mathematics.

After that, I'd say Algebraic Number Theory with JP Serre's textbook: A Course in Arithmetic.

I see from your edit that you found the Chevalley-Warning theorem. But if you are still interested in getting more detailed information about the solutions, it seems that there are some interesting regularities. If your quadratic form is nondegenerate (in the sense defined below) then it appears that there are always exactly p^2 solutions, including the zero solution.

First of all, the book of Jean-Pierre Serre, A Course of Arithmetic treats these matters in considerable detail, including a proof of the Chevalley-Warning theorem (page 16). But he also goes on to prove a classical result (page 34, due to Gauss maybe?), which shows that by a linear change of variables any quadratic form is equivalent to one with no off-diagonal terms. So you can reduce

ax^2 + by^2 + cz^2 + 2(exy + fxz + gyz) to a'x'^2 + b'y'^2 + c'z'^2

It is possible that the quadratic form is degenerate in the sense that one or more of a',b',c' turn out to be zero. However if none of the a', b' and c' are zero mod p, then it appears from my empirical tests that there are always p^2 solutions. You can experiment with this yourself if you have access to Mathematica. For example,

a = 1;
b = 1;
c = 1;
p = 7;
Clear[x, y, z]
solns = Solve[a x^2 + b y^2 + c z^2 == 0, {x, y, z}, Modulus -> p];
Length[solns]

The number of solutions, including the zero solution is 49 as claimed. I've tried a number of different prime moduli and various non-zero values for a, b, and c, and always gotten p^2

I haven't thought too much about how to prove all of this, but I thought you might still be interested.

p.s. What sort of imbecile would downvote a real mathematics post like this?

> This seems very confusing to me, as it is defining p-adic expansion of numbers in terms of p-adic numbers ...

This is just a hand-wavy, intuitive explanation of what p-adic numbers look like. The fact is that once you formalize everything about the [p-adic valuation](http://en.wikipedia.org/wiki/P-adic_valuation) and the p-adic numbers, it turns out that every p-adic number has the series expansion that you mentioned.

> For instance, why, in the p-adic world, are positive powers of p small, and negative powers large? It seems like a prime number to a large power would be large, no?

When dealing with p-adic numbers, you have to forget all your intuition about the usual notions of absolute values and ordering of the real numbers, since they don't apply. Everything in the p-adic world is based on the p-adic valuations, which give their own topologies and notions of size. The p-adic topologies are very different from the topology on R. For example, any point within an open ball in the p-adic numbers can be considered that ball's center. Quirky things like this make it initially hard to grasp the concepts of p-adic numbers and their associated arithmetic, but once you practice working with them enough, they start to make sense.

> How does the limit of the sequence that they're talking about equal 1/3?

This again has to do with the fact that convergence in the p-adic topology is different from convergence in the usual Euclidean topology.

Some good resources for learning more about p-adic numbers are the following:

1. Gouvêa, Fernando Quadros, p-adic Numbers: An Introduction (Amazon, SpringerLink)
   
2. Koblitz, Neal, p-adic Numbers, p-adic Analysis, and Zeta-Functions (Amazon, SpringerLink)
   
3. Robert, Alain M, A Course in p-adic Analysis (Amazon, SpringerLink)
   
4. Serre, Jean-Pierre, A Course in Arithmetic (Amazon, SpringerLink)
   
   For me personally, learning general valuation theory was very useful for understanding p-adic numbers.

I see how the title could be misleading. One of the most famous examples is Serre's A Course In Arithmetic