fundamental theorem of arithmetic

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21: 4.44 Other Applications

§4.44 Other Applications

… ►For applications of generalized exponentials and generalized logarithms to computer arithmetic see §3.1(iv). …

22: 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. …

23: 27.14 Unrestricted Partitions

… ►A fundamental problem studies the number of ways $n$ can be written as a sum of positive integers $\le n$, that is, the number of solutions of … ►Euler’s pentagonal number theorem states that …

24: 4.48 Software

… ►All scientific programming languages, libraries, and systems support computation of at least some of the elementary functions in standard floating-point arithmetic (§3.1(i)). … ►Here we provide links to the research literature describing the implementation of algorithms in software for the evaluation of functions described in this chapter when the arithmetic is nonstandard. … ►A more complete list of available software for computing these functions is found in the Software Index; again, software that uses only standard floating-point arithmetic is excluded. … ►

§4.48(ii) Interval Arithmetic

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25: 1.2 Elementary Algebra

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Binomial Theorem

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Arithmetic Progression

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§1.2(iv) Means

►The arithmetic mean of $n$ numbers $a_1,a_2,\mathrm{\dots },a_n$ is ►

1.2.17 $$A=\frac{a_1+a_2+\mathrm{\cdots }+a_n}{n}.$$

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Defines:

$A$: arithmetic mean (locally)

Symbols:

$n$: nonnegative integer

A&S Ref:

3.1.11

Permalink:

http://dlmf.nist.gov/1.2.E17

Encodings:

pMML, png, TeX

See also:

Annotations for §1.2(iv), §1.2 and Ch.1

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26: 3.12 Mathematical Constants

… ►The fundamental constant …

27: 8.24 Physical Applications

… ►With more general values of $p$, $E_p\left(x\right)$ supplies fundamental auxiliary functions that are used in the computation of molecular electronic integrals in quantum chemistry (Harris (2002), Shavitt (1963)), and also wave acoustics of overlapping sound beams (Ding (2000)).

28: Bibliography

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Characterization theorems for orthogonal polynomials. In Orthogonal Polynomials (Columbus, OH, 1989), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., Vol. 294, pp. 1–24.

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External Links:

Link, MathReview (Walter Van Assche), ZentralBlatt

Referenced by:

§18.3.

Encodings:

BibTeX

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Gauss, Landen, Ramanujan, the arithmetic-geometric mean, ellipses, $\pi$, and the Ladies Diary. Amer. Math. Monthly 95 (7), pp. 585–608.

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External Links:

ISSN 0002-9890, FullText, Document, MathReview (R. A. Askey), ZentralBlatt

Referenced by:

§19.9(i).

Encodings:

BibTeX

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The MasPar MP-1 as a computer arithmetic laboratory. J. Res. Nat. Inst. Stand. Technol. 101 (2), pp. 165–174.

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External Links:

Document

Referenced by:

§3.1(iv).

Encodings:

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Theorems on generalized Dedekind sums. Pacific J. Math. 2 (1), pp. 1–9.

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External Links:

MathReview (N. G. de Bruijn), ZentralBlatt

Referenced by:

(25.11.41), (25.11.42), §25.11(xii).

Encodings:

BibTeX

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A Centennial History of the Prime Number Theorem. In Number Theory, Trends Math., pp. 1–14.

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External Links:

MathReview (Giovanni Coppola), ZentralBlatt

Referenced by:

(25.16.2), (25.16.4), §25.16(i), §25.16.

Encodings:

BibTeX

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29: 16.21 Differential Equation

… ►A fundamental set of solutions of (16.21.1) is given by …For other fundamental sets see Erdélyi et al. (1953a, §5.4) and Marichev (1984).

30: 24.17 Mathematical Applications

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§24.17(iii) Number Theory

►Bernoulli and Euler numbers and polynomials occur in: number theory via (24.4.7), (24.4.8), and other identities involving sums of powers; the Riemann zeta function and $L$-series (§25.15, Apostol (1976), and Ireland and Rosen (1990)); arithmetic of cyclotomic fields and the classical theory of Fermat’s last theorem (Ribenboim (1979) and Washington (1997)); $p$-adic analysis (Koblitz (1984, Chapter 2)). …