Descriptions of areas/courses in number theory
- The ABC Conjecture
- 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
- Historical comments by David Masser and Joseph Oesterlé on the origins of the ABC conjecture
- The ABC-conjecture, Frits Beukers, ABC-day, Leiden, September 9, 2005 (fritsABCpresentation.pdf)
- 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)
- Arithmetical Geometry
- 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
- Topics in Arithmetical Geometry (Outline of a course by Shou-Wu Zhang - pdf)
- Grothendieck's SGA
- Lecture notes on the local Langlands correspondence (Michael Harris)
- Modular Mahler Measures, slides by Fernando Villegas
- Diophantine geometry in characteristic p: a survey, (Felipe Voloch)
- Draft of PCMI Lecture Notes on Open Questions in Arithmetic Algebraic Geometry (Alice Silverberg) (ps file 803K)
- Fermat's Last Theorem
- 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
- Computing irregular primes
- 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
- Charles Daney's pages on Fermat's last Theorem
- 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
- 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
- Pell equations
- Primes and factoring
- Progress made on de Polignac's conjecture
- Conjectures involving primes and quadratic forms, Zhi-Wei Sun
- 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; also see note on P.X. Gallagher's result on the singular series by Kevin Ford
- 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
- Small gaps between primes, transparencies of a talk by Jimena Sivak-Fischler
- Prime constellation records (Jens Kruse Andersen)
- Primzahlbeweise
- 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)
- 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;
Gordon Spence's Mersenne prime discovery
- Will Edgington's Mersenne Primes Page
- AMS Notices article on Primality Testing (Richard Pinch)
- Frobenius Pseudoprimes (Jon Grantham)
- Primes of the form k·2n+1 (Ray Ballinger and Wilfred Keller)
- Primes of the form k·2n-1 (Wilfred Keller)
- Hans Riesel's problem (Ray Ballinger and Wilfred Keller)
- Sierpinski's problem (Ray Ballinger and Wilfred Keller)
- Seventeen or Bust, a distributed attack on the Sierpinski problem, Seventeen Or Bust wiki
- Fermat number factoring status (Wilfred Keller)
- 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)
- The largest known composite Fermat number (John Cosgrave)
- Lucas-Lehmer criterion (Paul Garrett)
- Lucas–Lehmer test for Mersenne numbers (Wikipedia)
- Factorizations of xy + yx for 1 < y < x < 101 (Andrey Kulsha)
- Links to factoring programs (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
- Sums of integer cubes
- Table of non-negative integral solutions of n=x3+y3+z3 (Hisanori Mishima)
- Twin primes
- Elliptic Curves
- Sage Days 22: Computing with Elliptic Curves
- Topics in Algebraic Geometry: Elliptic Curves, Lecture course by Franz Lemmermeyer
- Honours Thesis and slidetalk on Descent on Elliptic Curves, Martin Leslie
- 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
- Student Projects (elliptic curves)
- Elliptic curves over Q (pdf) (Steve Finch)
- 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)
- An Elementary Introduction to Elliptic Curves (ps), Leonard S. Charlap, David P. Robbins, 1988, 1990
- 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)
- Draft chapters of elliptic curve handbook ECH1
- 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
- The Modular curves X0(N), Lecture notes by Bas Edixhoven
- Elliptic curves and right triangles (slides by Karl Rubin)
- Algebraic Number Theory
- The Sato-Tate Conjecture (Julian Rosen and Ralf Schmidt)
- Cubic fields (Wikipedia)
- Units and class groups in number theory and algebraic geometry, Serge Lang, Bull. Amer. Math. Soc. 6 (1982), 253-316
- Learning Algebraic Number Theory (Sam Ruth)
- 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 Juliana Belding
- 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 A. 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
- Richard Mollin's Quadratic Fields Research Area
- 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
- The Cyclotomic Fields Virtual Study Group
- 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
- 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)
- Modular forms over SL2(Q) (pdf) (Steve Finch)
- 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
- Ken Ribet's Modular Forms and Hecke Operators Course (Notes by 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
- Analytic Number Theory
- 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
- The Number Field Sieve
- 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
- Well-known constants
- Diophantine Approximation, Diophantine equations, Geometry of Numbers, Irrationality
- 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)
- Mordell's review, Siegel's letter to Mordell, Diophantine geometry and 20th century mathematics, Serge Lang, Gazette des Mathématiciens - n°63, January 1995, SMF
- Various diophantine equations (Seiji Tomota)
- 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)
- Table of approximations to π = 3.14159··· of the form a/b2 (Ismael Jiménez Calvo)
- Second thoughts on some topics from Diophantine approximation and analytic number theory (Cameron Stewart) - item 12 of "Publications in refereed journals and books"
- 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
- Nombres irrationels, nombres transcendants, Édouard Lebeau, Journal de Maths des Élèves 3, 1995
- Gisbert Wüstholz: Ausgewählte Kapitel der Zahlentheorie und der Geometrie, Vorläufige unvollständige Version, WS 1995/96
- 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
- Lattices in Cryptography and Cryptanalysis (Daniele Micciancio)
- The lattice challenge, TU Darmstadt
- Tree of primitive Pythagorean triples
- Quadratic reciprocity
- Squarefree numbers
- Abundant numbers
- Aliquot sequences, Perfect, Amicable numbers
- Bell Numbers
- Bernoulli Numbers
- Congruences
- Continued Fractions
- 3x+1
- Valuations
- Primitive roots and Artin's conjecture
- Fibonacci numbers
- Function fields
- Goldbach's Conjecture
- Hall's Conjecture
- Kurepa's Conjecture
- The Riemann Hypothesis
- 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); also see 70 % de RH
- Abundant Numbers and the Riemann Hypothesis (Keith Briggs, Experimental Math. 15, Issue 2 (2006), 251-256)
- SIMUW 2007: What is Riemann's Hypothesis? (A course for high school students)
- 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)
- Further systematic computations on the summatory function of the Möbius function, Tadej Kotnik, Jan van de Lune
- 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 zetafunctions (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 zetafunction and its relatives (Frits Beukers)
- The Riemann Hypothesis in a Nutshell (Glen Pugh)
- Xrays of the Riemann Zeta and Xi functions, James M. Hill, Robert K. Wilson
- Distribution Modulo 1
- Number Theory and Cryptography
- Introductory Number Theory
- Extremal Functions in Fourier Analysis
- Descriptions of Interesting Courses
- Lehmer's Problem:
- Markov numbers:
- 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
- Binomial Coefficients: Andrew Granville's Page
- Equal Sums of Like Powers/Tarry Escott Problem:Computing minimal equal sums of like powers
- Integer Sequences, Well-known Functions
- Selberg's Eigenvalue Conjecture:, Peter Sarnak, Notices of the AMS, November 1995
- Number Theory Bibliography: Joseph Eschgfäller at Ferrara
- Erdös-Strauss conjecture
- 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 compex 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
- Partitions
- Problems
- Combinatorial number theory
- 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)
- Ben Green's notes
- Carmichael numbers
- Carmichael's conjecture
- Catalan's conjecture
- Class number
Various other numbers
- Recreations in Number Theory
Return to Menu page
Return to Number Theory Web page
Last modified 20th May 2013