Reddit mentions of Number Theory: A Lively Introduction with Proofs, Applications, and Stories

Here's my rough list of textbook recommendations. There are a ton of Dover paperbacks that I didn't put on here, since they're not as widely used, but they are really great and really cheap.

Amazon search for Dover Books on mathematics

There's also this great list of undergraduate books in math that has become sort of famous: https://www.ocf.berkeley.edu/~abhishek/chicmath.htm

Pre-Calculus / Problem-Solving

• The Art of Problem Solving - Lehoczky/Ruczyk - in two volumes, both have full solutions available separately
  
  Calculus
  
  
• Calculus - Stewart - ignore the bad reviews, it's very clear and covers all of basic calculus well - it's just the book that most intro students use and as a consequence gets a bad rap
  
  
• Calculus - Spivak - this will take your understanding of calculus to the next level
  
  
• Calculus - Apostol - there are two volumes and they are very comprehensive, albeit difficult
  
  Linear Algebra
  
  
• Linear Algebra - Lay
  
  
• Linear Algebra Done Right - Axler
  
  Differential Equations
  
  
• Elementary Differential Equations - Boyce/DePrima
  
  
• Ordinary Differential Equations - Tenenbaum
  
  Number Theory
  
  
• Number Theory - Pommersheim for a change of pace and proof skills
  
  Proof-Writing
  
  
• Transition to Higher Mathematics - Dumas/McCarthy - helps to bridge the gap between "lower" and higher math
  
  Analysis
  
  
• The Way of Analysis - Strichartz
  
• Principles of Mathematical Analysis - Rudin
  
• Real and Complex Analysis - Rudin
  
  Complex Analysis
  
  
• Complex Analysis - Ahlfors
  
  Functional Analysis
  
  
• Functional Analysis - Rudin
  
  
• Introductory Functional Analysis with Applications - Kreyszig
  
  Partial Differential Equations
  
  
• Partial Differential Equations for Scientists and Engineers - Farlow
  
  Higher-dimensional Calculus and Differential Geometry
  
  
• Calculus on Manifolds - Spivak - for a modern take on multivariable calculus
  
  
• Differential Geometry - Kreyszig
  
  Abstract Algebra
  
  
• A First Course in Abstract Algebra - Fraleigh
  
• Abtract Algebra - Dummit and Foote
  
  Geometry
  
  
• The Elements - Euclid - for culture and for understanding where math has come from, plus it's still an awesome book after all these years; don't need to go through all of it
  
• Euclidean and Non-Euclidean Geometries - Greenberg
  
  Topology
  
  
• Introduction to Topology - Mendelson
  
  
• Topology - Munkres
  
  Set Theory and Logic
  
  
• Introduction to Set Theory - Hrbacek/Jech
  
  
• Elements of Set Theory - Enderton
  
  
• Mathematical Logic - Kleene
  
  
• Set Theory and the Continuum Hypothesis - Cohen
  
  Combinatorics / Discrete Math
  
  
• Combinatorics - Cameron
  
  
• Concrete Math - Graham/Knuth/Patashnik
  
  
• Discrete Math - Biggs
  
  Graph Theory
  
  
• A First Course in Graph Theory - Chartrand/Zhang
  
  P. S., if you Google search any of the topics above, you are likely to find many resources. You can find a lot of lecture notes by searching, say, "real analysis lecture notes filetype:pdf site:.edu"
  

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.