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Difference between revisions of "Number theory"

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'''Number theory''' is the field of [[mathematics]] associated with studying the [[integers]].
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'''Number theory''' is the field of [[mathematics]] associated with studying the properties and identities of [[ integer]]s.  
  
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==Overview==
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Number theory is a broad topic, and may cover many diverse subtopics, such as:
 +
*[[Modular arithmetic]]
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*[[Prime number]]s
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Some branches of number theory may only deal with a certain subset of the real numbers, such as [[integer]]s, [[positive]] numbers, [[natural number]]s, [[rational number]]s, etc. Some [[algebra]]ic topics such as [[Diophantine]] equations as well as some theorems concerning integer manipulation (like the [[Chicken McNugget Theorem ]]) are sometimes considered number theory.
  
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== Introductory Topics ==
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== Student Guides to Number Theory ==
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The following topics make a good introduction to number theory.
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* '''[[Number theory/Introduction | Introductory topics in number theory]]'''
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* [[Prime number | Primes]]
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** Covers different kinds of integers such as [[prime number]]s, [[composite number]]s,  [[perfect square]]s and their relationships ([[multiple|multiples]], [[divisor|divisors]], and more).  Also includes [[base number]]s and [[modular arithmetic]].
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** [[Sieve of Eratosthenes]]
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* '''[[Number theory/Intermediate | Intermediate topics in number theory]]'''
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** [[Prime factorization]]
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* '''[[Number theory/Olympiad | Olympiad topics in number theory]]'''
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* [[Composite number | Composite numbers]]
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* '''[[Number theory/Advanced topics | Advanced topics in number theory]]'''
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* [[Divisibility]]
 
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** [[Divisor]]s
 
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*** [[Common divisor]]s
 
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**** [[Greatest common divisor]]s
 
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*** [[Counting divisors]]
 
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** [[Multiples]]
 
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*** [[Common multiple]]s
 
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**** [[Least common multiple]]s
 
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* [[Division Theorem]] (the Division Algorithm)
 
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* [[Base numbers]]
 
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* [[Diophantine equation | Diophantine equations]]
 
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** [[Simon's Favorite Factoring Trick]]
 
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* [[Modular arithmetic]]
 
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** [[Linear congruence]]
 
  
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=== Introductory Resources ===
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== Resources ==
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==== Books ====
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=== Books ===
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* The AoPS [http://www.artofproblemsolving.com/Books/AoPS_B_Item.php?page_id=10 Introduction to Number Theory] by [[Mathew Crawford]].
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* Introductory
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==== Classes ====
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** ''the Art of Problem Solving Introduction to Number Theory'' by [[Mathew Crawford]] [https://artofproblemsolving.com/store/book/intro-number-theory (details)]
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* [[Introduction to Number Theory Course]] [http://www.artofproblemsolving.com/Classes/AoPS_C_ClassesS.php#begnum Details]
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** ''Elementary Number Theory: A Problem Oriented Approach '' by [[Joe Roberts]] [http://www.amazon.com/exec/obidos/ASIN/0262680289 (details)] Out of print but if you can find it in a library or used, you might love it and learn a lot. Writen caligraphically by the author.
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* General Interest
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== Intermediate Topics ==
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** ''Fermat's Enigma'' by Simon Singh [http://www.amazon.com/exec/obidos/ASIN/0385493622/artofproblems-20 (details)]
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An intermediate level of study involves many of the topics of introductory number theory, but involves an infusion of [[mathematical problem solving]] as well as [[algebra]].
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** ''Music of the Primes'' by Marcus du Sautoy [http://www.amazon.com/exec/obidos/ASIN/0066210704/artofproblems-20 (details)]
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* [[Diophantine equations]]
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** ''104 Number Theory Problems'' by Titu Andreescu, Dorin Andrica, Zuming Feng
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** [[Pell equation | Pell equations]]
 
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** [[Simon's Favorite Factoring Trick]]
 
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* [[Euclidean algorithm]]
 
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* [[Modular arithmetic]]
 
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** [[Linear congruence]]
 
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*** [[Chinese Remainder Theorem]]
 
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** [[Euler's Totient Theorem]]
 
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** [[Fermat's Little Theorem]]
 
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** [[Wilson's Theorem]]
 
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=== Intermediate Resources ===
 
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==== Classes ====
 
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*[[Intermediate Number Theory]] [http://www.artofproblemsolving.com/Classes/AoPS_C_ClassesS.php#intermnum (Details)]
 
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== Olympiad Topics ==
 
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An Olympiad level of study involves familiarity with intermediate topics to a high level, a few new topics, and a highly developed [[proof writing]] ability.
 
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* [[Diophantine equations]]
 
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** [[Pell equation | Pell equations]]
 
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** [[Simon's Favorite Factoring Trick]]
 
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* [[Modular arithmetic]]
 
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** [[Linear congruence]]
 
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*** [[Chinese Remainder Theorem]]
 
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** [[Euler's Totient Theorem]]
 
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** [[Fermat's Little Theorem]]
 
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** [[Wilson's Theorem]]
 
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** [[Quadratic reciprocity]]
 
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=== Olympiad Resources ===
 
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==== Books ====
 
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* [http://www.artofproblemsolving.com/Resources/Papers/SatoNT.pdf Number Theory by Naoki Sato]
 
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== Advanced Topics in Number Theory ==
 
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=== Algebraic Number Theory ===
 
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[[Algebraic number theory]] studies number theory from the perspective of [[abstract algebra]]. In particular, heavy use is made of [[ring theory]] and [[Galois theory]]. Algebraic methods are particularly well-suited to studying properties of individual prime numbers. From an algebraic perspective, number theory can perhaps best be described as the study of <math>\mathrm{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})</math>. Famous problems in algebraic number theory include the [[Birch and Swinnerson-Dyer Conjecture]] and [[Fermat's Last Theorem]].
 
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=== Analytic Number Theory ===
 
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[[Analytic number theory]] studies number theory from the perspective of [[calculus]], and in particular [[real analysis]] and [[complex analysis]]. The techniques of [[analysis]] and [[calculus]] are particularly well-suited to studying large-scale properties of prime numbers. The most famous problem in analytic number theory is the [[Riemann Hypothesis]].
 
  
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=== Elliptic Curves and Modular Forms ===
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=== E-Book ===
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(I don't really feel like writing this right now. Any volunteers?)
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* [https://www.math.muni.cz/~bulik/vyuka/pen-20070711.pdf ''Problems in Elementary Number Theory'' by Hojoo Lee]
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* [https://numbertheoryguy.com/publications/olympiad-number-theory-book/ ''Intermediate Number Theory'' by Justin Stevens]
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* [http://artofproblemsolving.com/articles/files/SatoNT.pdf ''Number Theory'' by Naoki Sato]
  
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=== Online Courses===
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*Introductory Number Theory
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** [https://thepuzzlr.com/courses/introduction-to-number-theory-course/ Introduction to Number Theory]
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* Intermediate
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** [https://artofproblemsolving.com/school/course/catalog/intermediate-numbertheory Intermediate Number Theory]
  
 
== Other Topics of Interest ==
 
== Other Topics of Interest ==
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=== Famous Unsolved Number Theory Problems ===
 
=== Famous Unsolved Number Theory Problems ===
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* [[Birch and Swinnerson-Dyer Conjecture]]
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* [[Birch and Swinnerton-Dyer conjecture]]
 
* [[Collatz Problem]]
 
* [[Collatz Problem]]
 
* [[Goldbach Conjecture]]
 
* [[Goldbach Conjecture]]
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* [[Twin Prime Conjecture]]
 
* [[Twin Prime Conjecture]]
  
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=== Books ===
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[[Category:Number theory]]
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Here are a list of general interest books:
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[[Category:Definition]]
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* ''Fermat's Enigma'' by Simon Singh [http://www.amazon.com/exec/obidos/ASIN/0385493622/artofproblems-20 (details)]
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[[Category:Mathematics]]
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* ''Music of the Primes'' by Marcus du Sautoy [http://www.amazon.com/exec/obidos/ASIN/0066210704/artofproblems-20 (details)]
 

Latest revision as of 22:08, 8 January 2024

Number theory is the field of mathematics associated with studying the properties and identities of integers.

Overview

Number theory is a broad topic, and may cover many diverse subtopics, such as:

Some branches of number theory may only deal with a certain subset of the real numbers, such as integers, positive numbers, natural numbers, rational numbers, etc. Some algebraic topics such as Diophantine equations as well as some theorems concerning integer manipulation (like the Chicken McNugget Theorem ) are sometimes considered number theory.

Student Guides to Number Theory

Resources

Books

  • Introductory
    • the Art of Problem Solving Introduction to Number Theory by Mathew Crawford (details)
    • Elementary Number Theory: A Problem Oriented Approach by Joe Roberts (details) Out of print but if you can find it in a library or used, you might love it and learn a lot. Writen caligraphically by the author.
  • General Interest
    • Fermat's Enigma by Simon Singh (details)
    • Music of the Primes by Marcus du Sautoy (details)
    • 104 Number Theory Problems by Titu Andreescu, Dorin Andrica, Zuming Feng

E-Book

Online Courses

Other Topics of Interest

These are other topics that aren't particularly important for competitions and problem solving, but are good to know.


Famous Unsolved Number Theory Problems

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