An introduction to the theory of numbers

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1. xzyyyzJan 2025
   Project Euler is collection of puzzles based on math. The coding and designs skills are completely secondary to the math. The algorithms (except for computational number theory, computation geometry) are completely secondary. So it is knowledge of math that is barrier.

   I keep learning math (and, sometimes, relatively new development in math) by solving problems. To start, I would recommend reading G.H. Hardy & E.M. Wright "An Introduction to the Theory of Numbers".

   With LeetCode -- this is more traditional exercise in coding. I did not pay much attention to it. So, sorry, cannot provide any advice

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2. tomatocracyDec 2024
   Absolutely - Hardy and Wright's An introduction to the theory of numbers is excellent but definitely doesn't need to be restricted to postgrads - its also fine for undergraduate level number theory (indeed it was a recommended textbook when I was an undergrad).

   It is quite dense but at least personally I find that style of textbook much more useful than the American style enormous textbooks which takes a chapter to explain what could be said in a paragraph. You just need to know to expect it will take you quite a while to read each page.

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3. tomatocracyMay 2022
   There are a couple of textbooks I haven't seen mentioned elsewhere in this thread which I found very useful as a first year undergraduate - they are concise in style but not to the point of being too difficult to follow (it's personal preference but I found I got on much better with this style of textbook):

   Davenport - The Higher Arithmetic

   Burkill - A First Course in Mathematical Analysis (and also A Second Course in Mathematical Analysis by Burkill and Burkill)

   Hardy and Wright - An introduction to the theory of numbers

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4. playing_coloursOct 2019
   Focusing on mathematics, I think about two kinds of foundational books.

   1. Historically foundational: books that brought important ideas, introduced a new branch, progressed the maths significantly. You can read them these days, but their content, proofs, language, etc. is not up to date. Examples: Elements by Euclid, Philosophiae Naturalis Principia Mathematica by Newton, Disquisitiones Arithmeticae by Gauss.

   2. Modern foundational books: The ones that can be used for studying now. Books that are clearly written, popular, cover their subject well. Examples: Visual Complex Analysis, Spivak’s Calculus, Linear Algebra Done Right, Hardy’s An Introduction to the Theory of Numbers.

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5. francassoMar 2018
   My own take on it:

   Lang's books about the basics I find lovely

   - Basic stuff: Basic Mathematics - Lang

   - One variable Calculus: A First Course in Calculus - Lang

   - Multi variable Calculus: Calculus of Several Variables - Lang

   - Linear Algebra: Linear Algebra - Lang

   - Number theory: An Introduction to the Theory of Numbers - Niven, Zuckerman, Montgomery

   Some more advanced stuff

   - Algebra: Algebra - Artin or Algebra, Chapter 0 - Aluffi

   - Complex Analysis: Functions of One Complex Variable I - Conway

   - Probability Basics: An Introduction to Probability Theory and its Applications I - Feller

   - Real Analysis and functional analysis basics: Real Analysis - Folland

   - Basic Differential Geometry: Elementary Differential Geometry - O'Neill

   - Riemann Surfaces (algebraic take): Algebraic Curves - Fulton

   - Differential Topology: Differential Topology - Guillemin, Pollack

   - Riemann Surfaces (analytic take): Compact Riemann Surfaces, an Introduction to Contemporary Mathematics - Jost

   - Modern Differential Geometry: Lectures on The Geometry of Manifolds - Nicolaescu

   - Functional analysis: Fundamentals of the Theory of Operator Algebras I - Kadison, Ringrose

   - Introduction to the Index Theory of which you actually have already seen some in the Riemann Surfaces books with the Riemann-Roch theorem: Index Theory with Applications to Mathematics and Physics - Bleecker, Doob Bavnbek

   - Homological Algebra: An Introduction to Homological Algebra - Weibel

   - Algebra for algebraic geometry: Commutative Ring Theory - Matsumura

   - Soft introduction to schemes: The Geometry of Schemes - Eisenbud, Harris

   - Algebraic geometry: Algebraic Geometry - Robin Hartshorne

   This should get you up more or less to what was current in the '60s :)

   Additional methodological note: I'm not suggesting going linearly through all these books. Well, perhaps going linearly thorough the basics would be a good idea, but after that I would follow my own interests.

   The important thing is really to have pen and paper and work things out by yourself, not just reading the book. I'm not saying you should try to prove all the theorems yourself or do all the exercises, that would take an unrealistic amount of time, but you can try to think about a theorem before reading its proof to see if you have a sense of which road is more likely to lead to a proof, and then try to reproduce the proof with pen and paper after having read it to check if you actually understood it.

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6. EpiMathDec 2017
   If you mean Hardy and Wright's Introduction to the Theory of Numbers, I agree it is excellent, but you can solve the great majority of the project Euler problems without going to quite that level. I particularly enjoyed Daniel Shank's Solved and Unsolved Problems in Number Theory but even that goes beyond what is necessary.
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