Recommended readings for PRIMES applicants and students

General

• Sandor Lehoczky and Richard Rusczyk, The Art of Problem Solving, Vol. 1: the Basics (AoPS, 2006); The Art of Problem Solving, Vol. 2: and Beyond (AoPS, 2006).
• Paul Zeitz, The Art and Craft of Problem Solving , 2nd ed. (Wiley, 2006).
• A Decade of the Berkeley Math Circle: The American Experience , edited by Zvezdelina Stankova and Tom Rike (American Mathematical Society, 2008).
• Evan Chen, The Napkin project

Linear Algebra

• Gilbert Strang, Introduction to Linear Algebra , 4th ed. (Wellesley-Cambridge Press, 2009).
• Video lectures by Prof. Gilbert Strang on MIT OpenCourseWare.
• More advanced reading (abstract linear algebra):
  • Sheldon Axler, Linear Algebra Done Right , 3rd ed. (Springer, 2015).
• Sergei Treil, Linear Algebra Done Wrong
• Isaiah Lankham, Bruno Nachtergaele and Anne Schilling, Linear Algebra - As an Introduction to Abstract Mathematics (Lecture Notes for MAT67, UC Davis, 2007).

Abstract Algebra

• Algebra Fact Sheet
• MATH E-222 Abstract Algebra , video lectures by Prof. Benedict Gross on Harvard Extension School’s Open Learning Initative.
• V.B. Alekseev, Abel’s Theorem in Problems and Solutions (Kluwer, 2004).
• Joseph Gallian, Contemporary Abstract Algebra , 7th ed. (Brooks Cole, 2009).
• More advanced reading:
  • Michael Artin, Algebra , 2nd ed. (Addison Wesley, 2010).

Number Theory

• Ivan Niven, Herbert S. Zuckerman, and Hugh L. Montgomery, An Introduction to the Theory of Numbers , 5th ed. (Wiley Text Books, 1991).
• Harold Davenport, The Higher Arithmetic: An Introduction to the Theory of Numbers , 7th ed. (Cambridge University Press, 1999).
• Lecture notes and assignments for the course 18.781, Theory of Numbers , by Prof. Abhinav Kumar, on MIT OpenCourseWare.

Combinatorics

• Miklos Bona, A Walk Through Combinatorics: An Introduction to Enumeration and Graph Theory , 2nd ed. (World Scientific, 2006).
• More advanced reading:
  • Richard P. Stanley, Enumerative Combinatorics , Vol. 1, 2nd ed. (Cambridge University Press, 2011).

Knot Theory

• Colin Adams, The Knot Book (American Mathematical Society, 2004).

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