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1: 24.1 Special Notation
Bernoulli Numbers and Polynomials
The origin of the notation B n , B n ( x ) , is not clear. …
Euler Numbers and Polynomials
Its coefficients were first studied in Euler (1755); they were called Euler numbers by Raabe in 1851. The notations E n , E n ( x ) , as defined in §24.2(ii), were used in Lucas (1891) and Nörlund (1924). …
2: 27.17 Other Applications
§27.17 Other Applications
Reed et al. (1990, pp. 458–470) describes a number-theoretic approach to Fourier analysis (called the arithmetic Fourier transform) that uses the Möbius inversion (27.5.7) to increase efficiency in computing coefficients of Fourier series. Congruences are used in constructing perpetual calendars, splicing telephone cables, scheduling round-robin tournaments, devising systematic methods for storing computer files, and generating pseudorandom numbers. … There are also applications of number theory in many diverse areas, including physics, biology, chemistry, communications, and art. …
3: 27.19 Methods of Computation: Factorization
Techniques for factorization of integers fall into three general classes: Deterministic algorithms, Type I probabilistic algorithms whose expected running time depends on the size of the smallest prime factor, and Type II probabilistic algorithms whose expected running time depends on the size of the number to be factored. … As of January 2009 the largest prime factors found by these methods are a 19-digit prime for Brent–Pollard rho, a 58-digit prime for Pollard p 1 , and a 67-digit prime for ecm. … These algorithms include the Continued Fraction Algorithm (cfrac), the Multiple Polynomial Quadratic Sieve (mpqs), the General Number Field Sieve (gnfs), and the Special Number Field Sieve (snfs). …The snfs can be applied only to numbers that are very close to a power of a very small base. The largest composite numbers that have been factored by other Type II probabilistic algorithms are a 63-digit integer by cfrac, a 135-digit integer by mpqs, and a 182-digit integer by gnfs. …
4: 27.18 Methods of Computation: Primes
§27.18 Methods of Computation: Primes
An overview of methods for precise counting of the number of primes not exceeding an arbitrary integer x is given in Crandall and Pomerance (2005, §3.7). …An analytic approach using a contour integral of the Riemann zeta function (§25.2(i)) is discussed in Borwein et al. (2000). … These algorithms are used for testing primality of Mersenne numbers, 2 n 1 , and Fermat numbers, 2 2 n + 1 . …
5: 26.11 Integer Partitions: Compositions
c ( n ) denotes the number of compositions of n , and c m ( n ) is the number of compositions into exactly m parts. c ( T , n ) is the number of compositions of n with no 1’s, where again T = { 2 , 3 , 4 , } . …
26.11.1 c ( 0 ) = c ( T , 0 ) = 1 .
The Fibonacci numbers are determined recursively by … Additional information on Fibonacci numbers can be found in Rosen et al. (2000, pp. 140–145).
6: 26.6 Other Lattice Path Numbers
§26.6 Other Lattice Path Numbers
Delannoy Number D ( m , n )
Motzkin Number M ( n )
Narayana Number N ( n , k )
§26.6(iv) Identities
7: 24.15 Related Sequences of Numbers
§24.15 Related Sequences of Numbers
§24.15(i) Genocchi Numbers
§24.15(ii) Tangent Numbers
§24.15(iii) Stirling Numbers
§24.15(iv) Fibonacci and Lucas Numbers
8: 26.5 Lattice Paths: Catalan Numbers
§26.5 Lattice Paths: Catalan Numbers
§26.5(i) Definitions
C ( n ) is the Catalan number. …
§26.5(ii) Generating Function
§26.5(iii) Recurrence Relations
9: 26.14 Permutations: Order Notation
As an example, 35247816 is an element of 𝔖 8 . The inversion number is the number of pairs of elements for which the larger element precedes the smaller: … The Eulerian number, denoted n k , is the number of permutations in 𝔖 n with exactly k descents. …The Eulerian number n k is equal to the number of permutations in 𝔖 n with exactly k excedances. …
§26.14(iii) Identities
10: 26.7 Set Partitions: Bell Numbers
§26.7 Set Partitions: Bell Numbers
§26.7(i) Definitions
§26.7(ii) Generating Function
§26.7(iii) Recurrence Relation
§26.7(iv) Asymptotic Approximation