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	ABOUT: [Introduction][History][Related areas][Subfields] POINTERS: [Texts][Software][Web links][Selected topics here] 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.

History

Applications and related fields

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).

Other web sites with this focus

• A good (and accessible) page for the Fibonacci numbers

Selected topics at this site

• Announcement of the sequence server at ATT
• The sum of the squares of the binomial coefficients
• Lucas theorem (computing binomial coefficients mod p and p-adically)
• Star of David theorem (LCM of sets of three binomial coefficients)
• Perfect squares among the binomial coefficients
• Integrality of some ratios of factorials
• The Catalan numbers -- some recurrence relations and other formulas.
• References on Catalan numbers
• Properties of Bernoulli numbers
• Clausen and von Staudt's theorem (denominators of Bernoulli numbers)
• The Bernoulli numbers and Bernoulli polynomials (to express the sum of the first n perfect k-th powers.)
• The Bell numbers (number of ways to partition n elements into nonempty subsets)
• The Stirling numbers (number of ways to partition n elements into k nonempty subsets)
• What are the Stirling numbers? (symmetric functions in {1,...,n})
• Stirling numbers and partial zeta sums
• Find Fibonacci numbers divisible by p^2 ?
• Fibonacci numbers mod p
• Divisors of Fibonacci numbers
• The Perrin sequence (3,0,2,..., a_(n+1)=a_(n-1)+a_(n-2).)
• Properties of the Perrin sequence, references
• Recurrence sequences with prime values at prime locations
• Topological proof of Van der Waerden's theorem (any finite partition of the set of natural numbers leaves an arithmetic progression in at least one subset).
• Add n to its "opposite"; repeat until a palindrome appears. Will this end if we start with 196? (open)
• Reverse the digits and add; get to a palindrome? (open)
• Asymptotics of a quadratic recurrence (a(n) = a(n-1)-a(n-1)^2 )
• f(x)=2x+1 implies f^n(x) prime for some n? (not necessarily)
• Iterate f(x) = 3x+1; always get a power of 2? (No)
• Length of chains in a dynamical system on Z (using operations add-one, subtract-one, and double)
• If S(a)=a^2 and T(a)=a+1, are there two words in S and T of equal length such that w1(a)=w2(a) for some integer a? No.
• Solve A_n = (n-1)A_{n-1} + (n-2)A_{n-2} in a closed form.
• String of 1000 digits which includes all numbers 1..1000 as substrings. (Why do we do these things?)
• Background and citations for the 1000 digits problem (deBruijn sequences)
• More about de Bruijn sequences (citations etc)
• Analyzing a Somos sequence (T_n=(T_(n-1) T_(n-4) + T_(n-2) T_(n-3) )/T_(n-5) ); fascinating connections with quasiperiodicity and the elliptic curve y^2+xy=x^3+x^2-2x.
• Maximal sets of integers with all subset sums distinct: Conway-Guy sequence