DLMF: Untitled Document

… ►Wagstaff (1978) gives complete prime factorizations of

N_{n}

and

E_{n}

for

n=20(2)60

and

n=8(2)42

, respectively. …

… ►where

p_{1},p_{2},\dots,p_{\nu\left(n\right)}

are the distinct prime factors of

n

, each exponent

a_{r}

is positive, and

\nu\left(n\right)

is the number of distinct primes dividing

n

. (

\nu\left(1\right)

is defined to be 0.) Euclid’s Elements (Euclid (1908, Book IX, Proposition 20)) gives an elegant proof that there are infinitely many primes. … ►(See Gauss (1863, Band II, pp. 437–477) and Legendre (1808, p. 394).) … ►

§27.2(ii) Tables

…

… ►

§6.16(ii) Number-Theoretic Significance of $\operatorname{li}\left(x\right)$

►If we assume Riemann’s hypothesis that all nonreal zeros of

\zeta\left(s\right)

have real part of

\tfrac{1}{2}

(§25.10(i)), then ►

6.16.5

\operatorname{li}\left(x\right)-\pi(x)=O\left(\sqrt{x}\ln x\right),

x\to\infty

,

ⓘ

Symbols:: $O\left(\NVar{x}\right)$ : order not exceeding, $\operatorname{li}\left(\NVar{x}\right)$ : logarithmic integral, $\ln\NVar{z}$ : principal branch of logarithm function, $x$ : real variable and $\pi(x)$ : number of primes $\leq x$
A&S Ref:: 5.1.50
Permalink:: http://dlmf.nist.gov/6.16.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §6.16(ii), §6.16 and Ch.6

►where

\pi(x)

is the number of primes less than or equal to

x

. … ►

See accompanying text — Figure 6.16.2: The logarithmic integral $\operatorname{li}\left(x\right)$ , together with vertical bars indicating the value of $\pi(x)$ for $x=10,20,\dots,1000$ . Magnify

§27.15 Chinese Remainder Theorem

… ►This theorem is employed to increase efficiency in calculating with large numbers by making use of smaller numbers in most of the calculation. …Choose four relatively prime moduli

m_{1},m_{2},m_{3}

, and

m_{4}

of five digits each, for example

2^{16}-3

,

2^{16}-1

,

2^{16}+1

, and

2^{16}+3

. Their product

m

has 20 digits, twice the number of digits in the data. …These numbers, in turn, are combined by the Chinese remainder theorem to obtain the final result

\pmod{m}

, which is correct to 20 digits. …

… ►

R. Chelluri, L. B. Richmond, and N. M. Temme (2000) Asymptotic estimates for generalized Stirling numbers. Analysis (Munich) 20 (1), pp. 1–13.

ⓘ

External Links:: ISSN 0174-4747, MathReview (F. T. Howard), ZentralBlatt
Referenced by:: §26.8(vii).
Encodings:: BibTeX

►

J. Chen (1966) On the representation of a large even integer as the sum of a prime and the product of at most two primes. Kexue Tongbao (Foreign Lang. Ed.) 17, pp. 385–386.

ⓘ

External Links:: MathReview (T. Hayashida)
Referenced by:: §27.13(ii).
Encodings:: BibTeX

… ►

M. Colman, A. Cuyt, and J. Van Deun (2011) Validated computation of certain hypergeometric functions. ACM Trans. Math. Software 38 (2), pp. Art. 11, 20.

ⓘ

External Links:: ISSN 0098-3500, Document, FullText, MathReview Entry, ZentralBlatt
Referenced by:: §13.29(v), §15.19(v).
Encodings:: BibTeX

… ►

M. D. Cooper, R. H. Jeppesen, and M. B. Johnson (1979) Coulomb effects in the Klein-Gordon equation for pions. Phys. Rev. C 20 (2), pp. 696–704.

ⓘ

External Links:: Document
Referenced by:: 5th item.
Encodings:: BibTeX

… ►

R. Crandall and C. Pomerance (2005) Prime Numbers: A Computational Perspective. 2nd edition, Springer-Verlag, New York.

ⓘ

External Links:: ISBN 978-0-387-25282-7; 0-387-25282-7, MathReview (Robert Juricevic), ZentralBlatt
Referenced by:: §27.12, §27.12, §27.18, §27.18, §27.18, §27.18, §27.18, §27.18, §27.19, §27.19, §27.19.
Encodings:: BibTeX

…

… ► … ►Then the number of plane partitions in

B(r,s,t)

is … ►The number of symmetric plane partitions in

B(r,r,t)

is … ►The number of cyclically symmetric plane partitions in

B(r,r,r)

is … ►The number of descending plane partitions in

B(r,r,r)

is …

… ►

p\left(\mathcal{D},n\right)