Descriptions of areas/courses in number theory

• Mathematics Subject Classification, 11-XX
• Eric Weisstein's World of Mathematics (Number Theory section)
• abc conjecture
• Abundant Numbers
• Aliquot Sequences, Perfect, Amicable Numbers
• Algebraic Number theory
• Analytic Number Theory
• Arithmetic Geometry
• Artin's Primitive Roots Conjecture
• Bell numbers
• Bernoulli numbers
• Carmichael numbers
• Carmichael's conjecture
• Catalan's conjecture
• Class number
• Combinatorial number theory
• 3x+1 (Collatz) Problem
• Computational Number Theory
• Congruent Numbers
• Well-known constants
• Continued Fractions
• Congruences
• Cryptography and Number Theory
• Diophantine Approximation, Diophantine equations, Geometry of Numbers, Irrationality
• Elliptic Curves
• Equal Sums of Like Powers: The Tarry-Escott Problem
• Exponential sums
• Extremal Functions in Fourier Analysis
• Fermat's Last Theorem
• Fibonacci numbers
• Function fields
• Goldbach's conjecture
• Integer Sequences
• Hall's conjecture
• Introductory Number Theory
• Irregular primes
• Kurepa's conjecture
• Lehmer's Problem
• Markov numbers
• L-functions, Modular Forms, Automorphic Forms
• Mahler measure
• Number Theory Biography (Joseph Eschgefäller)
• p-adic numbers
• Partitions
• Pell Equations
• Practical Numbers
• Pseudoprimes
• Primes and factoring
• Prime gaps
• Prime number theorem
• Problems
• PV Numbers
• Quadratic forms
• Quadratic reciprocity
• Riemann Hypothesis
• Riemann Hypothesis for elliptic curves over finite fields
• Selberg's Eigenvalue Conjecture
• Sieve theory
• Squarefree numbers
• Sums of integer cubes
• Twin primes
• Recreations in Number Theory
• Various numbers
• Wolstenholme's theorem
• Videos

The ABC Conjecture

• Why abc is still a conjecture (Peter Scholze and Jacob Stix)
• Mathematical proof that rocked number theory will be published (Nature, April 3, 2020)
• Titans of Mathematics Clash Over Epic Proof of ABC Conjecture (Qantamagazine, September 2018)
• Notes on the Oxford IUT workshop by Brian Conrad
• An ABC proof too tough even for mathematicians, Kevin Hartnett Boston Globe, November 4, 2012
• The abc conjecture, as easy as 1, 2, 3 ⋯ or not, Alex Ghitza, The Conversation, 26 November 2012
• The ABC's of Number Theory (Noam Elkies)
• Reken mee met ABC (Bart de Smit, Gillien Geuze), Nieuw Archief voor Wiskunde (5th series) 8 (2007), 26-30, Reken Mee met ABC
• The ABC-conjecture, Frits Beukers, ABC-day, Leiden, September 9, 2005
• ABC@home (finding abc triples related to the ABC conjecture)
• Special day on the ABC-conjecture, Intercity Number Theory Seminar, September 9th 2005
• It's As Easy As abc, Andrew Granville and Thomas J. Tucker, AMS Notices, November 2002, 49
• The ABC Conjecture Home Page (Abderrahmane Nitaj)

Riemann hypothesis for elliptic curves over finite fields

• The history of the zeta-function in characteristic p (Peter Roquette)
• The Riemann Hypothesis for elliptic curves, Jasbir Chahal, Brian Osserman
• The Riemann hypothesis over finite fields: from Weil to the present day, James Milne
• Early history of the Riemann Hypothesis in positive characteristic, Franz Oort and Norbert Schappacher, ALM 35, Higher Education Press and International Press, 2016, pp. 595-631
• Curves over finite fields, Richard Griffon

Arithmetic Geometry

• The origins of Langlands' conjectures, a slide-talk by William Casselman
• A p-adic approach to rational points on curves, Bjorn Poonen, Bull. Amer. Math. Soc. September 2020
• Quanta Magazine article on the Langlands correspondence, April 6, 2020
• VaNTAGe : a new virtual seminar on open conjectures in number theory and arithmetic geometry
  • Sato-Tate distributions (Lecture notes from the 2016 Arizona Winter School) Andrew Sutherland
• Afterword to the article Arithmetic on curves (Barry Mazur) BAMS 55, 353-358, 2018
• Simons Collaboration on Arithmetic Geometry, Number Theory, and Computation
• Modular Arithmetic: Driven by Inherent Beauty and Human Curiosity, Richard Taylor
• The Weil conjectures (Wikipedia)
• Langlands Program, trace formulas, and their geometrization, Edward Frenkel, Bull. Amer. Math. Soc. October 12, 2012
• Arithmetic on curves, Barry Mazur, Bull. Amer. Math. Soc. 14 (1986), 207-259
• The l-adic revolution in number theory, a video of a talk by Nick Katz at the IHES Colloquium in honour of Alexander Grothendieck, January 12, 2009
• Conferences in Arithmetic Geometry (Kiran Kedlaya)
• Videos of lectures, Clay Mathematics Institute 2006 Summer School on Arithmetic Geometry, July 17-August 11, 2006, Mathematisches Institut, Georg-August-Universität, Göttingen
• Rational points on curves (Shou-Wu Zhang)
• Grothendieck's SGA
• Lecture notes on the local Langlands correspondence (Michael Harris)
• Modular Mahler Measures, slides by Fernando Villegas
• Course notes in number theory (Felipe Voloch)
• Draft of PCMI Lecture Notes on Open Questions in Arithmetic Algebraic Geometry (Alice Silverberg) (ps file 803K)

Fermat's Last Theorem

• Remarks on the History of Fermat's Last Theorem - Michael Rosen, 16 February 2016
• Quanta magazine article on recent progress on the Langlands programs
• Ken Ribet's 2017 lecture on Fermat's Last Theorem
• Fermat's Last Theorem, a BBC Horizon programme by Simon Singh, Review of BBC's Horizon program, "Fermat's Last Theorem" (Andrew Granville) Notices of the AMS, January 1997
• The Way to the Proof of Fermat's Last Theorem Gerhard Frey, 2009
• Fermat's Last Theorem and Elliptic Curves - Gerhard Frey
• The generalized Fermat equation Frits Beukers
• Fermat's Last Theorem (Wikibooks)
• The proof of Fermat's Last Theorem by R. Taylor and A. Wiles, Gerd Faltings, Notices of the AMS, 42, July 1995
• Beal's conjecture
• A proof of the full Shimura-Taniyama-Weil conjecture is announced, Henri Darmon, Notices of the AMS, December 1999
• A report on Wiles' Cambridge lectures, K. Rubin, A. Silverberg, Bull. Amer. Math. Soc. 31 (1994), 15-38
• William Hammond's Fermat Archive
• Notes on Fermat's Last Theorem, A.J. van der Poorten, Canadian Mathematical Society Series of Monographs and Advanced Texts, Wiley-Interscience, January, 1996, ISBN 0-471-06261-8
  • Review of Alf van der Poorten's Notes on Fermat's Last Theorem by Andrew Granville
• The Solving of Fermat's Last Theorem, Karl Rubin, Ohio State University Distinguished Lecture May 22, 1997
• Le Théorème de Fermat (Karim Belabas and Catherine Goldstein)

Irregular primes

• Irregular primes to 163 million, Joe Buhler and David Harvey
• Large-scale verification of Vandiver's conjecture (slides by David Harvey)

Pell equations

• Michel Waldschmidt's lecture notes on L'équation dite de Pell-Fermat and Équations Diophantiennes
• Pell equation bibliography, 1658-1943 (Franz Lemmermeyer)
• Linear Recurrences for Pell Equations (John Robertson)
• Størmer's theorem
• Pell's equation (Andrej Nowicki)
• Pell's equation (MacTutor History of Mathematics Archive)
• Record-Holder Solutions of Pell's Equation (Tomás Oliveira e Silva)
• Solving the Pell equation, H.W. Lenstra Jr., Notices of the AMS, February 2002

Pseudoprimes

• Baillie–PSW probabilistic primality test (Wikipedia)
• Lucas pseudoprimes (Robert Baillie and Samuel Wagstaff Jr.)
• Lucas pseudoprimes (Robert Baillie and Samuel Wagstaff Jr.)
• Pseudoprimes (Wikipedia)

Primes and factoring

• Graduate student Jared Duker Lichtman proves prime number conjecture (Quanta magazine June 6, 2022)
• Mathematicians Outwit Hidden Number Conspiracy, Quanta magazine, January 3, 2022
• James Maynard discusses his proof that infinite primes exist missing each base 10 digit - he uses 7 as his arbitrary example
• Kaisa Matomäki Dreams of Primes, Quanta Magazine, July 20, 2017
• Prime-Wiki
• James Maynard: Small gaps between primes (lecture held within the framework of the Hausdorff Trimester Program: Harmonic Analysis and Partial Differential Equations and the Workshop: Analytic Number Theory of the Hausdorff Center for Mathematics 17.07.2014)
• Conjectures on Representations Involving Primes, Zhi-Wei Sun
• Mathematicians Discover Prime Conspiracy
• Andrica's conjecture
• Prime Numbers: A Much Needed Gap Is Finally Found, John Friedlander, AMS Notices, June-July 2015
• Polymath article on recent advances in the study of prime numbers
• Primes in intervals of bounded length, Andrew Granville, Bull. Amer. Math. Soc. 52 (2015), 171-222
• Primo, a primality proving program based on the ECPP algorithm (Marcel Martin)
• Problems on combinatorial properties of primes (Zhi-Wei Sun)
• Super Twin Prime Conjecture, a message to the Number Theory List, Feb. 6, 2014, from Zhi-Wei Sun
• Factorizations of bⁿ±1, b=2,3,5,6,7,10,11,12 Up to High Powers, third Edition, free download from AMS
• Firoozbakht's conjecture
• Slide talk by Jonathan Sondow: Lerch quotients and primes, Fermat-Wilson quotients, and the Wieferich-non-Wilson prime 14771; also Gel'fond's power tower conjecture
• Landau's problems on primes (János Pintz)
• Expository article on the recent theorem of Goldston, Pintz, and Yildirim on small gaps between prime numbers, K. Soundararajan
• The Challenge of Large Numbers (Richard Crandall)
• The ECMNET Project
• Small gaps between prime numbers: The work of Goldston-Pintz-Yildirim, K. Soundararajan, Bulletin Amer. Math. Soc. 44 (2007), 1-18
• Prime constellation records (Jens Kruse Andersen)
• Primes of the form x²+ny² (Marios Magioladitis)
• The Green-Tao Theorem on arithmetic progressions in the primes: an ergodic point of view, Bryna Kra, Bull. Amer. Math. Soc. 43 (2006), 3-23
• About the cover: On the distribution of primes-Gauss' tables, Yuri Tschinkel, Bull. Amer. Math. Soc. 43 (2006), 89-91
• The Niven Lectures, (Carl Pomerance) March 21-23, 2005, University of British Columbia
• It is easy to determine whether a given integer is prime, Andrew Granville, Bull. Amer. Math. Soc. 42 (2005), 3-38
• Generalized Fermat Primes Search
• PRIMES is in P- a non-specialist account by Folkmar Bornemann
• Exposition of the Primes is in P theorem (Kevin Ford)
  • Primality Testing for Beginners, Lasse Rempe-Gillen, Rebecca Waldecker, Student Mathematical Library Vol. 70, AMS 2014
• Review of Prime numbers: A computational perspective, Reviewer: Jeremy Teitelbaum
• The Prime Pages
• The Nth prime page
• Largest Known Primes
• Mersenne Primes: History, Theorems and Lists
• MacTutor History of Mathematics: Prime Numbers
• Mersenne Prime Search;
• AMS Notices article on Primality Testing (Richard Pinch)
• Appendix 1: Factorization results (Hisanori Mishima)
• Factorizations of Cyclotomic Numbers (Mitsuo Morimoto)
• Factorization of Generalized Repunits (Andy Steward)
• All known factors of 30+ digits found by Pollard's p-1 method (Andy Steward)
• Repunit primes and factors of 10ⁿ±1 (Torbjörn Granlund)
• Surprising connections between prime numbers and physics (Matthew Watkins)
• Lucas-Lehmer criterion (Paul Garrett)
• Lucas-Lehmer test for Mersenne numbers (Wikipedia)
• Factorizations of x^y + y^x for 1 < y < x < 101 (Andrey Kulsha)
• Arithmetic Progression of 26 primes has been found

Prime number theorem

• Complex Variables, by Robert Ash and W.P. Novinger has a chapter on the prime number theorem

Prime gaps

• Slides from Oberwolfach lecture series "Sieve theory and small gaps between primes" on the Polymath8 project on prime gaps
  • Slide 1
  • Slide 2
  • Slide 3
• L. E. Dickson's conjecture on prime k-tuples (The Prime Pages)
• Some conjectures on the gaps between consecutive primes (Marek Wolf)
• k-tuples (Thomas Engelsma)

Sums of integer cubes

• Table of non-negative integral solutions of n=x³+y³+z³ (Hisanori Mishima)

Twin primes

• Three lectures by Terry Tao on bounded gaps between primes, IHES, 2015
• Together and Alone, Closing the Prime Gap, Erica Klarreich
• Lemoine's conjecture
• Introduction to Twin Primes and Brun's Constant (Pascal Sebah)
• Pentium FDIV bug, (Thomas Nicely, the Pentium bug and the twin prime constant)
• Enumeration of the twin primes-Brun's constant to 10¹⁴ (Thomas Nicely)
• Primzahlzwillingsrekorde (K.-H. Indlekofer)

Elliptic Curves

• Quanta magazine article on elliptic curves
• 18.783 Elliptic Curves (MIT, Spring 2019) (Andrew Sutherland)
• Rational points on, and the arithmetic of, elliptic curves: A tale of two books (and an article), J.H. Silverman, Bull. Amer. Math. Soc. January 5, 2017
• How many rational points does a random curve have?, Wei Ho, Bull. Amer. Math. Soc. 51 (2014), 27-52
• Sage Days 22: Computing with Elliptic Curves
• Topics in Algebraic Geometry: Elliptic Curves, Lecture course by Franz Lemmermeyer
• Lang-Trotter revisited, Nicholas M. Katz, Bull. Amer. Math. Soc. 46 (2009), 413-457
• Finding meaning in error terms, Barry Mazur, Bulletin of the AMS, 45 (2008), 185-228
• Ribet-Stein notes on Serre's conjecture (pdf)
• A normal form for elliptic curves, Harold Edwards, Bull. Amer. Math. Soc. 44 (2007), 393-422
• Math 583 at University of Washington: The Birch and Swinnerton-Dyer Conjecture, lecture course (William Stein)
• Millennium Prize: the Birch and Swinnerton-Dyer Conjecture, Daniel Delbourgo, The Conversation
• Average ranks of elliptic curves, Baur Bektemirov, Barry Mazur, William Stein and Mark Watkins
• DEA 2003/04: Elliptic functions and elliptic curves, lecture notes by Jan Nekovář
• E.R. Hedrick Lectures of the MAA, August 2003: Rational points on modular elliptic curves (Henri Darmon)
• X₀(11) and X₁(11), the Euler system of Heegner points (Tom Weston)
• Ranks of elliptic curves, K. Rubin, A. Silverberg, Bull. AMS. 39, 2002, 455-474
• Review of Euler Systems, Reviewer: Henri Darmon
• The arithmetic of elliptic curves and diophantine equations (Loic Merel)
• Computing the rank of an elliptic curve, Undergraduate thesis, Jeff Achter, Brown University 1992
• Hyperelliptic curves allowing fast arithmetic (Tanya Lange)
• Elliptic curve handbook ECH1 (Ian Connell)
• History of elliptic curves rank records (Andrej Dujella)
• High rank elliptic curves with prescribed torsion (Andrej Dujella)
• Course Notes by Jim Milne: Algebraic number theory, Class field theory, Algebraic Geometry, Elliptic Curves, Modular functions and forms, Abelian varieties, Etale Cohomology
• Seminar Notes on Elliptic Curves and Formal Groups: J. Lubin, J.-P. Serre and J. Tate, Summer Institute on Algebraic Geometry, Woods Hole, 1964
• Seminar notes: Aspects of complex multiplication (Don Zagier), An introduction to group schemes (René Schoof), notes by John Voight
• Math 574, (An introduction to rational points on elliptic curves through examples of interesting elliptic curves), Jerrold Tunnell
• Efficient Verification of Tunnell's Criterion, Eric Bach and Nathan C. Ryan
• Elliptic curves and right triangles (slides by Karl Rubin)

Algebraic Number Theory

• Iwasawa theory, an introduction by Romyar Sharifi, AMS Notices, January 2019
• Lectures and videos by John Coates, David Loeffler and Sarah Zerbes, Romyar Sharifi, Christopher Skinner and Ralph Greenberg, Arizona Winter School 2018: Iwasawa Theory
• Representation theory and number theory, notes of lectures by Benedict Gross in 2011
• The Sato-Tate Conjecture (Julian Rosen and Ralf Schmidt)
• On the history of the Sato-Tate conjecture (Tetsushi Ito)
• Cubic fields (Wikipedia)
• Units and class groups in number theory and algebraic geometry, Serge Lang, Bull. Amer. Math. Soc. 6 (1982), 253-316
• Rademacher Lectures: 2009-2010, John Coates, (Iwasawa theory, Cyclotomic Iwasawa theory, The general Main Conjecture, The Tate-Shafarevich group and Iwasawa theory)
• Computing the Hilbert Class Polynomial Using p-adic Lifting, slide talk by Reinier Bröker
• Dedekind zeta function, notes by Ben Brubaker
• Expository notes on algebraic number theory, eg. Kummer's Lemma (Keith Conrad)
• On a theorem of Jordan, J.-P. Serre, Bull. Amer. Math. Soc. 40 (2003) 429-440
• The idelic approach to number theory, introduction to local fields, the modular curves X₀(11) and X₁(11) (Tom Weston)
• Oberwolfach: Explicit Algebraic Number Theory, Notes by John Voight
• How many fields share a common discriminant? (Daniel Mayer)
• Cubic number Fields (Daniel C. Mayer)
• Review of Cohomology of Number Fields, J. Neukirch, A. Schmidt, K. Wingberg (Reviewer: F.Q. Gouvêa) Bulletin AMS 39, 101-107, 2002
• Galois modules in arithmetic, by Boas Erez
• MAS4002: Algebraic Number Theory, Course notes by Robin Chapman, University of Exeter
• Survey of Euclidean Number Fields (ps file 371K) (Franz Lemmermeyer)
• Quadratische Zahlkörper, Lecture notes by Franz Lemmermeyer
• Dan Bernstein's Math 514
• Course Note: Algebraic number theory, Class field theory, Algebraic Geometry, Elliptic Curves, Modular functions and forms, Abelian varieties, Etale Cohomology
• Binary Cubic Forms and Cubic Number Fields
• Course Notes for elementary and algebraic number theory, Ivan Fesenko
• The Nonabelian Reciprocity Law for Local Fields, Jonathan Rogawski, Notices AMS, Vol 47, 2000
• Algebraic Number Theory and commutative algebra, lecture notes by Robert Ash
• Math 254B (Number Theory), lecture notes on class field theory, abelian extensions of number fields etc (Kiran Kedlaya)
• Algebraic Number Theory and Automorphic L-functions, lecture notes by Ching-Li Chai

L-functions, Modular Forms, Automorphic Forms

• The L-Functions and Modular Forms Database, John E. Cremona, John W. Jones, Andrew V. Sutherland, and John Voight, AMS Notices October, 2021
• Overconvergent modular forms and their explicit arithmetic, Jan Vonk, Bull. Amer. Math. Soc. 58 (2021), 313-356
• Lecture videos in number theory posted by Ikuya Kaneko
• Singular moduli for real quadratic fields, Jan Vonk, IAS Princeton video
• L-functions (Kevin Buzzard)
• The 1-2-3 of modular forms reviewed by Amanda Folsom, Bull. Amer. Math. Soc. 46 (2009), 527-533
• Uncovering a New L-function, Andrew R. Booker, Notices of the AMS, Volume 55, October 2008
• A directory of all known L-functions (Matthew Watkins)
• Review by Stephen Gelbart of Advanced Analytic Number Theory: L-Functions, Carlos Moreno
• Artin L-Functions: A Historical Approach, (Noah Snyder)
• The principle of functoriality, James Arthur, BAMS 40 39-53, 2003
• Review of Arithmeticity in the theory of automorphic forms, Reviewer: Hiroyuki Yoshida
• Math 574 - A Graduate course in automorphic forms and representations (Stephen Miller)
• Database of Automorphic L-functions (Stephen Miller)
• The idelic approach to number theory, introduction to local fields, the modular curves X₀(11) and X₁(11), L-functions and cyclotomic units, the Euler system of Heegner points (expository papers by Tom Weston)
• Modular forms (Igor Dolgachev)
• Modular forms database, by William Stein
• L-functions and cyclotomic units (Tom Weston)
• Paul Garrett
  • Vignettes on Automorphic and Modular Forms, Representation, L-functions and Number Theory
  • Bibliography for Automorphic and Modular Forms, L-functions and Representation Theory
• Modular Forms and Hecke Operators Course (Ken Ribet and William A. Stein)
• Some Old Problems and New Results about Quadratic Forms, W. Duke, Notices of the AMS, February 1997
• L-functions of the Selberg class S
• Galois representations and modular forms, Kenneth A. Ribet, Bull. Amer. Math. Soc. 32 (1995), 375-402

Mahler measure

• Talks on Mahler measure (Matilde Lalín)
• The many aspects of Mahler's measure

Analytic Number Theory

• The TME-EMT Project (Fully explicit results in multiplicative number theory are often scattered through the literature. The aim of this site is to present annotated bibliographies in order to keep track of the current knowledge.)
• Waring’s Problem: A Survey, R.C. Vaughan and T.D. Wooley
• Dirichlet's divisor problem, James Cann
• The role of the Ramanujan conjecture in analytic number theory, Valentin Blomer and Farrell Brumley, Bull. Amer. Math. Soc. January 2013
• Additive combinatorics reviewed by Ben Green, Bull. Amer. Math. Soc. 46 (2009), 489-497
• Vinogradov's three-primes theorem, notes by Timothy Gowers (dvi 54K)
• Analytic Number Theory and Applications: Collection of papers on the occasion of the 60th birthday of Anatolli Alexeevich Karatsuba, Proc. Steklov Institute 218 (1997)
• Lecture notes for Math 259: Introduction to Analytic Number Theory (Spring 1998) (Noam Elkies)

Computational Number Theory

• Algorithms in algebraic number theory, H. W. Lenstra, Bull. Amer. Math. Soc. 26 (1992), 211-244
• Sage Days 16: UPC Barcelona, Spain - Computational Number Theory, June 22-27, 2009 (transcripts and videos of talks including Experimental methods in number theory and analysis by Henri Cohen)
• NIST Digital Library of Mathematical Functions (Preview release)
• Distributed search for Fermat number divisors
• Future directions in algorithmic number theory (American Institute of Mathematics)
• Factorization of F10
• Some Number Records (Paul Zimmermann)
• Computational class field theory, A course given at the Middle East Technical University, Ankara, 1997 by Henri Cohen (dvi 255K)
• FactorWorld (Scott Contini)
• The NFSNET Project (a distributed effort using the Number Field Sieve to factor large numbers of the form bⁿ±1)
• Computing the Ramanujan Tau function (Denis Xavier Charles)
• A new solution to the equation τ(p) ≡ 0 (mod p) (Nik Lygeros and Olivier Rozier)
• Ramanujan's tau function τ(n) for n up to 1,000,000 (Chris Smyth)

Congruent Numbers

• A Trillion Triangles (report on recent record-breaking work on finding all congruent numbers up to 10¹²
• Congruent number problem (Keith Conrad)
• The Congruent Number Problem (William Stein)
• Right Triangles and Elliptic Curves (Karl Rubin)
• Congruent numbers: Table of right-angled triangles (Allan MacLeod)

Well-known constants

• Euler's constant: Euler's work and modern developments, J. C. Lagarias, Bull. Amer. Math. Soc, 19th July 2013
• De Bruijn-Newman constant (Wikipedia)
• Mathematical Constants, Steve Finch, CUP 2003
• A prime-representing function, Bull. Amer. Math. Soc., Volume 53, Number 12 (1947), 1196-1196 and Mills' constant (PlanetMath)

Diophantine Approximation, Diophantine equations, Geometry of Numbers, Irrationality

• Quanta magazine article on the recent proof of Schinzel-Zassenhaus conjecture by Vesselin Dimitrov
• A short course by Yuri Bilu on the recent proof of Schinzel-Zassenhaus conjecture by Vesselin Dimitrov (SPARC Lecture series)
• Computational Number Theory topics (Seiji Tomita)
• The sum-of-three-cubes for the number 3 has been solved by Andrew Booker and Andrew Sutherland
• What is a Diophantine m-tuple? (Andrej Dujella)
• Freiburg und die Kreiszahl Pi (Dieter Wolke 2007)
• Explicit Methods for Solving Diophantine Equations, Henri Cohen, Arizona winter School, Tuscon 2006
• Apéry's theorem and problems for the values of Riemann's zeta function and their q-analogues, a D. Sc. thesis by Wadim Zudilin
• Effective methods for Diophantine equations (Florian Luca)
• Notes on transcendental number theory (Math 249A, 2010), K. Soundararajan
• Old and new conjectured diophantine inequalities, Serge Lang, Bull. Amer. Math. Soc. 23 (1990), 37-75
• Diophantine approximations, Diophantine equations, transcendence and applications, T.N. Shorey
• Michel Waldschmidt's lecture notes on L'équation dite de Pell-Fermat and Équations Diophantiennes
• Lattice point problems (Paul Scott)
• Serge Lang's review of Mordell's book Diophantine Equations. Also see Serge Lang's commentary on Mordell and Siegel
• Lectures from AWS 2008: Special Functions and Transcendence in pdf and video format
• Undecidability in Number Theory, Bjorn Poonen, Notices AMS 55, 2008
• An introduction to irrationality and transcendence methods, course and project outline, draft lecture notes for lectures 1, 2, 3, 5, Arizona Winter School 2008, Michel Waldschmidt
• Introduction to Diophantine methods: irrationality and transcendence, course notes by Michel Waldschmidt
• Algebraic and Transcendental Numbers (Stéphane Fischler)
• Questions d'irrationalité (et de transcendance): hier et aujourd'hui, (Michel Waldschmidt)
• papers on irrationality of certain constants (Stéphane Fischler
• Hilbert's tenth problem page
• Criteria for irrationality of Euler's constant (Jonathan Sondow)
• Multiple Zeta Values and Euler-Zagier Numbers, etc. (Michel Waldschmidt)
• Zeta values on the Web (Wadim Zudilin)
• Diophantine m-tuples (Andrej Dujella)
• A bibliography of papers related to simultaneous diophantine approximation (Keith Briggs)
• Linear Independence Measures for Logarithms of Algebraic Numbers, (Cetraro lectures of Michel Waldschmidt, July 2000)
• Online integer relations interface (CECM, Simon Fraser University)
• Perfect Lattices (Jacques Martinet and Christian Batut)
• Review of J. Martinet's Perfect lattices in Euclidean spaces, by Gabriele Nebe
• Le Journal de Maths des Élèves
• Thue equations (Clemens Heuberger)
• Publications of Benne de Weger
• A new extreme abc-example (Benne de Weger)
• Diophantine Approximations, Mathematical Transactions 2 (1996) A Collection of papers dedicated to the memory of Prof. N.I. Feldman
• The lattice challenge, TU Darmstadt
• Tree of primitive Pythagorean triples
• The Cakravāla method for solving quadratic diophantine equations (M.D. Srinivas)

Quadratic reciprocity

• Reciprocity laws and Galois representations: recent breakthroughs, Jared Weinstein, Bull. Amer. Math. Soc. 53 (2016), 1-39
• Wouter Castryck's proof of quadratic reciprocity (pdf)
• Review of Franz Lemmermeyer's book Reciprocity laws, from Euler to Eisenstein, by Özlem Imamoglu, Bull. Amer. Math. Soc., 44 (2007), 647-652
• Webliography on Reciprocity Laws (Franz Lemmermeyer)
• Résidus quadratiques, Lois de réciprocité quadratique (Cyril Banderier)

Squarefree numbers

• First occurrences of square-free gaps and an algorithm for their computation (Louis Marmet)

Abundant numbers

• Abundant numbers (Wolfram MathWorld)
• Superabundant numbers and the Riemann hypothesis, Amir Akbary and Zachary Friggstad, The American Mathematical Monthly, 116 (3) (2009), 273-275

Aliquot sequences, Perfect, Amicable numbers

• Qanta Magazine article on odd perfect numbers
• Odd perfect numbers have at least ten distinct prime factors (Pace Nielsen)
• Perfect numbers - an elementary introduction, John Voight
• Largest prime factor of an odd perfect number (Takeshi Goto)
• Aliquot sequences page (Christophe Clavier)
• An account of perfect numbers, MacTutor History of Mathematics
• Wolfgang Creyaufmüller's page on aliquot sequences
• Aliquot sequences (Wieb Bosma)
• Aliquot sequences, (Juan L. Varona)
• Multiply Perfect Numbers (Achim Flammenkamp)
• Recent papers on odd perfect numbers and triperfect numbers (Andrew C. Palfreyman)

Bell Numbers

• Aurifeuillian factorization and the period of the Bell numbers modulo a prime (Peter Montgomery, Sangil Nahm, Samuel Wagstaff Jr.)

Bernoulli Numbers

• The Bernoulli Number Page (Bernd C. Kellner)
• Introduction to Bernoulli and Euler polynomials (Zhi-Wei Sun) (pdf file)
• A Bibliography of Bernoulli Numbers (Karl Dilcher and Ilja Slavutskii)
• Computing Bernoulli and Tangent Numbers (Richard Brent)
• Old and new algorithms for computing Bernoulli numbers (David Harvey)

Congruences

• RSA algorithm (Gihan Marasingha)
• The chinese remainder theorem

Continued Fractions

• Continued fractions, Michel Waldschmidt
• Continued fractions, a talk by John Barrow
• Continued fractions and r/j-algorithm, V.I. Shmoylov, Rostov-on-Don: Publishing House of the Southern Federal University, 2012, 608 p.
• Continued fractions and summation of series, V.I. Shmoylov, Ya.S. Korovin, D.Ya. Ivanov, Rostov-on-Don: SFU Publishing House, 2018
• Solution of Systems of Linear Algebraic Equations by Continued Fractions, Vladimir Shmoilov, Ya. C. Korovin, Southern Federal University, Rostov-on-Don, USSR, 2017 (16Mb pdf)
• Continued fractions: The Bibliography, Vladimir Shmoilov, Vladimir Voitulevich, Southern Federal University, Rostov-on-Don, USSR, 2017
• Continued fractions in local fields and nested automorphisms, (slides by Antonino Leonardis)
• Harold Stark's explanation of some exotic continued fractions of cubic irrationalities
• Continued fractions in quadratic fields (Alf van der Poorten)
• Three Connections to Continued Fractions, (Ezra Brown)
• Continued fractions and modular functions, William Duke, Bull. Amer. Math. Soc. 42 (2005), 137-162
• Lecture on continued fractions (Pavel Guerzhoy - pdf)
• Continued fractions and modular forms, Seminar by Ilan Vardi (summary by Cyril Banderier)
• Continued Fractions, an introduction by Alexander Bogomolny
• Continued fractions (Adam van Tuyl)
• Continued fractions from Euclid to the present day, preprint by I. Vardi, P. Flagolet, B. Vallée
• Continued fractions and factoring (Niels Lauritzen)

3x+1

• The Simplest Math Problem No One Can Solve, a Veritasium video
• Some progress on the 3x+1 problem due to Terry Tao
• The Collatz conjecture, Littlewood-Offord theory, and powers of 2 and 3 (Terry Tao)
• Jeff Lagarias' Article on the 3x+1 Problem and its Generalizations
• Annotated bibliography of the 3x+1 problem, by Jeff Lagarias
• Computational verification of the 3x+1 conjecture by Tomás Oliveira e Silva
• Translation of the Collatz letter (to Professor Mays) (Marion Meudt and John Read)

Primitive roots and Artin's conjecture

• Artin's Primitive Root Conjecture - A Survey, Pieter Moree
• Least primitive roots page, by Tomás Oliveira e Silva
• Our primitive roots (Chris Lyons)

Fibonacci and Tribonacci numbers

• Tribonacci numbers (Mathworld)
• Congruences for Fibonacci numbers (Zhi-Hong Sun, pdf)

Function fields

• Review of Michael Rosen's Number theory in function fields, Bull. AMS 41 (2004), 127-133 (David Goss)

Goldbach's Conjecture

• Proof by Harald Helfgott of the weak Goldbach conjecture
• Unification of Goldbach's conjecture and the twin prime conjecture, a message to Number Theory List, January 29, 2014 from Zhi-Wei Sun
• Mixed sums of primes and other terms (Zhi-Wei Sun)
• Goldbach conjecture numerical verification results by Tomás Oliveira e Silva
• Verifying Goldbach's Conjecture up to 4 x 10¹⁴ (Jörg Richstein)
• Goldbach variations: problems with prime numbers, Alessandro Zaccagnini
• Goldbach Conjecture research (Mark Herkommer)

Hall's Conjecture

• Hall's conjecture (Ismael Jiménez Calvo)

Kurepa's Conjecture

• Yves Gallot on !p (mod p)
• On Guy's problem B44: Kurepa's Conjecture

The Riemann Hypothesis

• Andrew Odlyzko: Correspondence about the origins of the Hilbert-Polya Conjecture
• Numerical observations on zeta and related functions, a project of Yuri Matiyasevich
• Finite Euler products (Yuri Matiyasevich)
• From Prime Numbers to Nuclear Physics and Beyond
• Approximation of Riemann's zeta function by finite Dirichlet series: multiprecision numerical approach (preprint, Gleb Beliakov, Yuri Matiyasevich)
• The Riemann hypothesis and eigenvalues of related Hankel matrices I (preprint, Yu. V. Matiyasevich)
• Three lectures on the Riemann zeta function (Steve Gonek)
• Zeroes of Riemann's zeta function on the critical line with 20000 decimal digits accuracy, Yuri Matiyasevich and Gleb Beliakov
• Calculation of Riemann's zeta function via interpolating determinants, preprint, Yuri Matiyasevich
• New conjectures about zeroes of Riemann's zeta function, Yuri Matiyasevich
• Alan Turing and Number Theory, Video by Yuri Matiyasevich, June 2012, Alan Turing Centenary Conference, Manchester
• Hidden Life of Riemann's Zeta Function, Yuri Matiyasevich
• An artless method for calculating approximate values of zeros of Riemann's zeta function, Yuri Matiyasevich
• Slide talk - New computations of the Riemann zeta function, Jonathan Bober (joint work with Ghaith Hiary)
• The Riemann Hypothesis is 150 years old
• A New Conjecture Related to the Riemann Hypothesis, Jack Good and Bob Churchhouse, 1968
• Voronin Universality Theorem (Jörn Steuding-MathWorld)
• Guy Robin's theorem on the Riemann hypothesis
• Ramanujan, Robin, the Riemann Hypothesis, and Recent Results, slidetalk by Jonathan Sondow, Ramanujan 125 Conference
• On Robin's criterion for the Riemann Hypothesis (YoungJu Choie, Nicholas Lichiardopol, Pieter Moree, Patrick Solé)
• Abundant Numbers and the Riemann Hypothesis (Keith Briggs, Experimental Math. 15, Issue 2 (2006), 251-256)
• What is Riemann's Hypothesis? (Barry Mazur and William Stein)
• Problems of the Millennium: The Riemann Hypothesis Peter Sarnak, 2004
• Turing and the Riemann Hypothesis, Andrew Booker, Notices of the AMS., 53 (2006) 1208-1211
• Zeroes of the Riemann zeta function and Dirichlet L-functions (Oruganti Shanker)
• Riemann's zeta function and beyond, Stephen Gelbart, Stephen Miller, Bull. AMS 41 (2004), 59-112
• The Riemann Hypothesis, B. Conrey, Notices of the AMS, 341-353, March 2003
• Directory of zeta functions (Matthew Watkins)
• Noncommutative Geometry, Trace Formulas and the Zeros of the Riemann Zeta Function, a course by Alain Connes
• Links to Riemann's original paper
• Introduction to the Riemann Zeta function (Xavier Gourdon and Pascal Sebah)
• Tables of zeros of the zeta function (Andrew Odlyzko)
• The Riemann zeta function and its relatives (Frits Beukers)
• Xrays of the Riemann Zeta and Xi functions, James M. Hill, Robert K. Wilson

Number Theory and Cryptography

• Lattices in Cryptography and Cryptanalysis (Daniele Micciancio)

Introductory Number Theory

• A Collection of Algebraic Identities, Titus Piezas III
• Lecture notes on elementary number theory (Bruce Ikenaga)
• Clay Mathematics Institute Introductory Workshop in Algorithmic Number Theory (MSRI Video Archive)
• An Index for G.H. Hardy and E.M Wright: An Introduction to the theory of numbers
• Introductory Number Theory 1 (Don Rideout)
• Discovering Number Theory, John Jones and Jeff Holt, W.H. Freeman,
• Math 780: Elementary Number Theory, Notes by Michael Filaseta, 1997
• Zahlentheorie (Notes by Winfried Bruns)

Extremal Functions in Fourier Analysis

• Notes by Miguel Lerma, (dvi 51K, ps 276K)

Lehmer's Problem

• Mathematician Measures the Repulsive Force Within Polynomials, Quanta magazine article on Vesselin Dimitrov's proof of the Schinzel-Zassenhaus conjecture
• Lehmer's Problem (Michael Mossinghoff)

Markov numbers

• Autour de l'équation de Markoff, Slides by Michel Waldschmidt, September 2009
• Markov numbers (Wikipedia)
• The Markov (unicity) conjecture for prime powers, Anitha Srinivasan (pdf) Also see (PlanetMath)

Practical numbers

• Giuseppe Melfi's Page on Practical Numbers

PV numbers (Pisot-Vijayaraghavan and Salem numbers)

• Review by M. Mendes France: Pisot and Salem numbers, M.J. Bertin, A. Decomps-Guilloux, M. Grandet-Hugot, M. Pathiaux-Delefosse, J. P. Schreiber, Birkhäuser 1992, ISBN 0-8176-2648-4, Bull. Amer. Math. Soc. 29 (1993) 274-278
• The arithmetic and geometry of Salem numbers, E. Ghate and E. Hironaka, Bulletin AMS 38 (2001), 293-314

Quadratic forms

• Lectures on quadratic forms, C. L. Siegel, TIFR 1967
• Ramanujan's ternary quadratic form (Wikipedia)
• Binary Quadratic Forms, Genus Theory, and Primes of the Form p = x² + ny², Josh Kaplan

Binomial Coefficients

• Andrew Granville's Page

Equal Sums of Like Powers/Tarry Escott Problem

• Chen Shuwen's Equal Sums of Like Powers Page
• Computing minimal equal sums of like powers

Integer Sequences, Well-known Functions

• The On-Line Encyclopedia of Integer Sequences (OEIS)
• Complete Annotated Bibliography of Work Related to Sidon Sequences (Kevin O'Bryant)
• Neil Sloane's Integer Sequences Page
• Eight hateful sequences (Neil Sloane)

Eigenvalue conjecture

• Selberg's Eigenvalue Conjecture:, Peter Sarnak, Notices of the AMS, November 1995

Number Theory Bibliography

• Joseph Eschgfäller at Ferrara

Exponential sums

• Exponential Sums and Differential Equations, N.M. Katz, Annals of Mathematics Studies 124, Princeton University Press 1990, Exponential sums over finite fields and differential equations over the complex numbers, Nicholas M. Katz Bull. Amer. Math. Soc. 23 (1990), 269-309
• Exponential sums, The estimate of Hasse-Davenport-Weil (Peter Roquette)
• The determination of Gauss sums, Bruce C. Berndt; Ronald J. Evans, Bull. Amer. Math. Soc. 5 (1981), 107-129

p-adic numbers

• Summer tutorial in p-adic analysis (Jack Thorne)
• p-adic numbers (Wikipedia)
• Review of A course in p-adic analysis, A.M. Robert (Reviewer N. Yui) Bulletin AMS 39, 113-119, 2002
• Lecture notes on p-adic numbers (Andrew Baker)

Partitions

• Addition and Counting: The Arithmetic of Partitions, S. Ahlgren, K. Ono, Notices of the AMS, Volume 48, October 2001

Problems

• Twin primes conjecture
• Polignac's conjecture
• Prime triplet conjecture
• k-tuples conjecture
• Riesel's problem
• Farhadian's conjecture
• Zhi-Wei Sun's square conjecture with prize
• The 24-conjecture with $2400 prize (Zhi-Wei Sun)
• 1-3-5-Conjecture and 1-2-3-Conjecture (Zhi-Wei Sun)
• Some mysterious representations of integers, a message to Number Theory Mailing List, October 25, 2015 (Zhi-Wei Sun)
• Problems in Elementary Number Theory (Peter Vandendriessche and Hojoo Lee)
• Some new problems in additive combinatorics, Zhi-Wei Sun
• List of conjectural series for powers of π and other constants (Zhi-Wei Sun)
• Open Conjectures on Congruences (Zhi-Wei Sun)
• Problems in number theory from the 2003 Arnold Ross summer program (Keith Conrad)
• West Coast Number Theory Problems
• Millennium Prize Problems
• Vsevolod Lev's Problem Page
• Some challenging unsolved problems in Number Theory (Hisanori Mishima)
• Unsolved problems in number theory, logic and cryptography (Tim Roberts)

Combinatorial number theory

• Conjectures involving arithmetical sequences, Zhi-Wei Sun
• Combinatorial Number Theory in China (Zhi-Wei Sun)
• Groups and Combinatorial Number Theory, a survey by Zhi-Wei Sun (pdf)
• Covering Systems, Restricted Sumsets, Zero-sum Problems and their Unification (Zhi-Wei Sun)
• Ergodic Prime Number Theorems Seminar, (supported by the Morningside Center)

Sieve theory

• What is the Parity Phenomenon?, John Friedlander and Henryk Iwaniec, Notices of the AMS, August 2009, Volume 56, 817-818
• Visualizing the Sieve of Eratosthenes, David N. Cox, AMS Notices, May 2008, 55
• A Tale of Two Sieves, C. Pomerance, AMS Notices, December 1996
• Sieves in number theory, by George Greaves, Reviewer H. Halberstam, Bull. AMS. 40, 2003, 109-119
• Sieve Methods, Masters Thesis, Denis Xavier Charles, SUNY Buffalo 2000 (pdf 501K)

Carmichael numbers

• Carmichael numbers and pseudoprimes (Richard Pinch)

Carmichael's conjecture

• Carmichael's Totient Function Conjecture (Eric W. Weisstein - Mathworld)

Catalan's conjecture

• Recent Work of Robert Styer and Reese Scott on the generalised Pillai equation
• A cyclotomic proof of Catalan's conjecture (Jeanine Daems)
• Catalan's Conjecture: Another old diophantine problem solved, Tauno Metsänkylä, Bull. AMS 41 (2004), 43-57
• Reflection, Bernoulli numbers and the proof of Catalan’s conjecture (Preda Mihăilescu)
  • Thaine's theorem (Wikipedia)
• Papers on Catalna's conjecture by Yuri Bilu
• The Prime Glossary: Catalan's problem
• Catalan's problem (Yu.V. Nesterenko )

Class number

• Class Number (Wolfram Mathworld)
• List of number fields with class number one (Wikipedia)
• Heegner's solution to the class number 1 problem
• The class number one problem for imaginary quadratic fields, Jeremy Booher
• Euler's famous prime generating polynomial and the class number of imaginary quadratic fields, Paulo Ribenboim, L'Enseignement Mathématique, 34 (1988), 24-42
• Quadratic Diophantine equations, the class number and the mass formula, Goro Shimura, Bull. AMS. 43 (2006), 285-304
• Computing in quadratic orders (John Robertson)
• Dorian Goldfeld: The Gauss class number problem
• Dorian Goldfeld: The Gauss class number problem, Bull. Amer. Math. Soc. 11, July 1985
• Table of class numbers of imaginary quadratic fields (Mark Watkins)

Various other numbers

• Stern-Brocot tree
• Stoneham numbers
• Perron numbers
• Cullen and Woodall numbers
• Cullen numbers (Wikipedia)
• Smith numbers (Shyam Sunder Gupta)
• Rare numbers(Shyam Sunder Gupta)
• Fascinating Factorials (Shyam Sunder Gupta)
• Amazing number 108 (Shyam Sunder Gupta)
• Harmonic numbers (Takeshi Goto)

Recreations in Number Theory

• Palindrome world records (Jason Doucette)
• Fascinating Triangular Numbers (Shyam Sunder Gupta)
• Elliptic curves in recreational number theory (Allan Macleod)
• Number Recreations (Shyam Sunder Gupta)

Wolstenholme's theorem

• Wolstenholme's theorem
• A proof of Wolstenholme's theorem (Mathematics Stack Exchange)

Videos

• Lecture videos in number theory (Ikuya Kaneko)
• Videos from BIRS Workshop 20w2254, Alberta Number Theory Days XII (Online), May 2-3, 2020, Banff
• Automorphic Forms and the Langlands Program, MSRI Summer Graduate School, 2017
• Sage Days 16: UPC Barcelona, Spain - Computational Number Theory, June 22-27, 2009 (transcripts and videos of talks including Experimental methods in number theory and analysis by Henri Cohen)
• The l-adic revolution in number theory, a video of a talk by Nick Katz at the IHES Colloquium in honour of Alexander Grothendieck, January 12, 2009
• Alan Turing and Number Theory, Video by Yuri Matiyasevich, June 2012, Alan Turing Centenary Conference, Manchester
• Counting Galois U₄(F_p)-extensions using Massey products, a video by Ján Mináč
• Singular moduli for real quadratic fields, Jan Vonk, IAS Princeton video
• Fermat's Last Theorem, a BBC Horizon programme by Simon Singh, Review of BBC's Horizon program, "Fermat's Last Theorem" (Andrew Granville) Notices of the AMS, January 1997
• Arizona Winter School
• Clay Mathematics Institute Introductory Workshop in Algorithmic Number Theory (MSRI Video Archive)
• Online Math Courses, videos and lectures from leading universities. This has links to some excellent number theory courses.
• The Life and Numbers of Richard Guy, a video by Hugh Williams at CNTA 2016
• Interview with Peter Swinnerton-Dyer (Alan Macfarlane)
• Interview with John Coates by Alan Macfarlane: part1, part2

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Last modified 15th July 2022