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	ABOUT: [Introduction][History][Related areas][Subfields] POINTERS: [Texts][Software][Web links][Selected topics here] 11J: Diophantine approximation, transcendental number theory

Introduction

History

Applications and related fields

see also 11K60

Subfields

• 11J04: Homogeneous approximation to one number
• 11J06: Markov and Lagrange spectra and generalizations
• 11J13: Simultaneous homogeneous approximation, linear forms
• 11J17: Approximation by numbers from a fixed field
• 11J20: Inhomogeneous linear forms
• 11J25: Diophantine inequalities, See also 11D75
• 11J54: Small fractional parts of polynomials and generalizations
• 11J61: Approximation in non-Archimedean valuations
• 11J68: Approximation to algebraic numbers
• 11J70: Continued fractions and generalizations, See also 11A55, 11K50
• 11J71: Distribution modulo one, See also 11K06
• 11J72: Irrationality; linear independence over a field
• 11J81: Transcendence (general theory)
• 11J82: Measures of irrationality and of transcendence
• 11J83: Metric theory
• 11J85: Algebraic independence; Gelfond's method
• 11J86: Linear forms in logarithms; Baker's method
• 11J89: Transcendence theory of elliptic and abelian functions
• 11J91: Transcendence theory of other special functions
• 11J93: Transcendence theory of Drinfel´d and t-modules [new in 2000]
• 11J95: Results involving abelian varieties [new in 2000]
• 11J97: Analogues of methods in Nevanlinna theory (work of Vojta et al.) [new in 2000]
• 11J99: None of the above but in this section

Parent field: 11: Number Theory

Browse all (old) classifications for this area at the AMS.

Textbooks, reference works, and tutorials

Classic references in this area include

• Baker, Alan: "Transcendental number theory", Cambridge University Press, Cambridge, 1990. 165 pp. ISBN 0-521-39791-X
• Lang, Serge: "Introduction to Diophantine approximations", Springer-Verlag, New York, 1995. 130 pp. ISBN 0-387-94456-7
• Schmidt, Wolfgang M.: "Diophantine approximation", Lecture Notes in Mathematics, 785. Springer, Berlin, 1980. 299 pp. ISBN 3-540-09762-7

See also the resources for Number theory in general.

Software and tables

Other web sites with this focus

Selected topics at this site

• Using integrals to show that pi isn't 22/7.
• Plenty of algorithms to compute pi.
• Programs to compute decimal digits of pi
• Sample algorithm to compute many digits of pi quickly.
• Bailey/Borwein/Plouffe method of computing digits of pi.
• Spigot algorithms to compute individual digits of pi.
• Computations of (individual) decimal digits of pi
• Citations to literature concerning pi.
• Irrationality of pi.
• Pointers to proofs that pi=3.14... is irrational
• Pi is irrational -- proof by Muse
• Summary of transcendental numbers; proof of transcendence of pi, e.
• Proof that Euler's constant e=2.718... is irrational
• Quality of approximations of an irrational by rationals.
• Fun with pi and e
• Irrationality of zeta(2) and zeta(3) and representation of zeta by integrals
• Basics on showing that a number is irrational.
• Explanations of various techniques for proving a number to be irrational
• Sqrt(n) is irrational if n is not a square: history and types of proof.
• Might Euler's constant gamma be rational? (unlikely)
• Copland-Erdos constant (a real number normal in all bases)
• Finding the best rational approximation to a real number.
• Mahler classification of transcendental numbers: how well can a number be approximated by algebraic numbers?
• Measures of quality of approximations by rationals (Farey, continued fractions)
• Maximizing a sum of sines (of different periods) (really a question of approximating a number by rationals).
• The S-unit equation: Solving X+Y=Z subject to X, Y, and Z having all their prime divisors in a fixed set S.
• Integer solutions to (2x+y)^n = 2(x+y)^n + y^n?
• Catalan's conjecture: that 8 and 9 are the only two consecutive positive perfect powers.
• Some (mostly) algebraic expressions which are nearly integral
• Books on Nevanlinna theory
• Algebraic independence of Gamma(1/4) from other constants
• Mahler function sum(x^(2^k))