Reddit mentions of A Classical Introduction to Modern Number Theory (Graduate Texts in Mathematics) (v. 84)

It's definitely upper graduate/PhD level. In an upper PhD level course labeled "Special Topics in Number Theory" at an Ivy League school, they might start expecting you to know it. At a more standard university, it would be something that the PhD students challenge themselves to work through together, or something special that an adviser would work with you on, not really something that people expect you to know or offer a big course in. It's too specialized and too long and any piece of it could be a whole course or two on it's own.

If you want to challenge yourself to eventually understand Fermat's Last Theorem, pick up a book in Number Theory (like this), keep on picking up new books in Number Theory and don't stop reading. I definitely think it's worth it, you can learn a lot of great math that way. And if you want to learn more math, grad school is always a good idea.

Well without knowing your background it's pretty hard to give you a recommendation... Ireland and Rosen is a classic. Note that this is a different book (and author) from /u/FunkMetalBass's comment.

Set Theory:

Naive Set Theory

Number Theory:

Elementary Number Theory

Introduction to Analytic Number Theory

A Classical Introduction to Modern Number Theory

Topology:

Topology

Introduction to Topological Manifolds

Liebeck's Concise Introduction to Pure Mathematics is a great text for introducing students to the basic tools required in abstract algebra, number theory and analysis, but doesn't go into great depth.

It's kind of a standard text but for abstract algebra I think Dummit and Foote is remarkably clear.

Ireland and Rosen's Classical Introduction to Modern Number Theory is a classic, but maybe more intermediate.

Elementary Number Theory by Jones is very good.

I would say that it would depend on the problem. If you cannot solve the first ten, I would be worried, as they can all be solved by simple brute force methods. I have a degree in Astrophysics, and some of the 300 and 400 problems are giving me pause, so if you are stuck there you are in good company.

There are elegant solutions to each problem, if you want to delve into them, but the first handful, the first ten especially, can be simply solved.

Once you get beyond the first ten or so, the mathematical difficulty ratchets up. There are exceptions to that rule of course, but by and large, it holds.

If you are interested in Number Theory, the best place to start is a number theory course at a local university. Mathematics, especially number theory, is difficult to learn by yourself, and a good instructor can expound, not only on the math, but also on the history of this fascinating subject.

Gauss, quite arguably the finest mathematician to ever live loved number theory; of it, he once said:

> Mathematics is the queen of sciences and number theory is the queen of mathematics. She often condescends to render service to astronomy and other natural sciences, but in all relations she is entitled to the first rank.

Although my personal favorite quote of his on the subject is:

> The enchanting charms of this sublime science reveal themselves in all their beauty only to those who have the courage to go deeply into it.

If you are interested in purchasing some books about number theory, here are a handful of recommendations:


Number Theory (Dover Books on Mathematics) by George E. Andrews


Number Theory: A Lively Introduction with Proofs, Applications, and Stories by James Pommersheim, Tim Marks, Erica Flapan


An Introduction to the Theory of Numbers by G. H. Hardy, Edward M. Wright, Andrew Wiles, Roger Heath-Brown, Joseph Silverman


Elementary Number Theory (Springer Undergraduate Mathematics Series) by Gareth A. Jones , Josephine M. Jones

and it's companion


A Classical Introduction to Modern Number Theory (Graduate Texts in Mathematics) (v. 84) by Kenneth Ireland, Michael Rosen

and a fun historical book:


Number Theory and Its History (Dover Books on Mathematics) Paperback by Oystein Ore

I would also recommend some books on

Markov Chains

Algebra

Prime number theory

The history of mathematics

and of course, Wikipedia has a good portal to number theory.

Rosen's Modern Number Theory (hands down) for number theory and Halmos' Finite-Dimensional Vector Spaces for linear algebra. There are other texts available for linear algebra but Halmos is a pretty solid option.