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11B: Sequences and Sets (including Fibonacci,...)

Introduction

This is largely the study of arithmetic properties of well-known sets of integers: binomial coefficients, Fibonacci numbers, and so on. More generally one looks at sequences defined recursively, say, and inquires about their congruence properties, primality, etc. (Questions about rate of growth, for example, are more likely considered in section 40: Sequences (of real numbers).

This area includes a number of Erdös-like topics: Covering congruences, additive bases for the integers, van der Waerden's theorem on arithmetic progressions, etc.

Subfields

• 11B05: Density, gaps, topology
• 11B13: Additive bases, See also 05B10
• 11B25: Arithmetic progressions, See also 11N13
• 11B34: Representation functions
• 11B37: Recurrences, For applications to special functions, See 33-XX
• 11B39: Fibonacci and Lucas numbers and polynomials and generalizations
• 11B50: Sequences (mod m)
• 11B57: Farey sequences; the sequences 1^k, 2^k, ...
• 11B65: Binomial coefficients; factorials; q-identities, See also 05A10, 05A30
• 11B68: Bernoulli and Euler numbers and polynomials
• 11B73: Bell and Stirling numbers
• 11B75: Other combinatorial number theory
• 11B83: Special sequences and polynomials
• 11B85: Automata sequences
• 11B99: 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

A nice survey of related results is in Erdös, P.; Graham, R. L.: "Old and new problems and results in combinatorial number theory", Université de Genève, L'Enseignement Mathématique, Geneva, 1980. 128pp.

"A primer for the Fibonacci numbers", edited by Marjorie Bicknell and Verner E. Hoggatt, Jr. The Fibonacci Association, San Jose State University, San Jose, Calif., 1972. 173 pp. MR50#12906

Bicknell, Marjorie: "A primer on the Pell sequence and related sequences", Fibonacci Quart. 13 (1975), no. 4, 345--349. MR52#8018

Some texts in combinatorics include quite a bit of "combinatorial number theory", including

• Stanley, Richard P., "Enumerative combinatorics", Wadsworth & Brooks/Cole Advanced Books & Software, Monterey, Calif., 1986. 306 pp. ISBN 0-534-06546-5
• Pomerance, Carl; Sárközy, András: "Combinatorial number theory"; Handbook of combinatorics, Vol. 1, 2, 967--1018, Elsevier, Amsterdam, 1995.

Online arithmetic properties of binomial coefficients [Andrew Granville]

See also the references for number theory in general.

Software and tables

• Sloane's sequence server (identifies an integer sequence from the first few terms).

Selected topics at this site

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Last modified 2000/01/14 by Dave Rusin. Mail: