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31: 27.16 Cryptography

§27.16 Cryptography

… ►The primes are kept secret but their product $n=pq$, an 800-digit number, is made public. …With the most efficient computer techniques devised to date (2010), factoring an 800-digit number may require billions of years on a single computer. For this reason, the codes are considered unbreakable, at least with the current state of knowledge on factoring large numbers. … ►Thus, $y\equiv x^r(modn)$ and . …

32: 27.21 Tables

§27.21 Tables

… ►Bressoud and Wagon (2000, pp. 103–104) supplies tables and graphs that compare $\pi \left(x\right),x/\ln x$, and $\mathrm{li}\left(x\right)$. … ►Lehmer (1941) gives a comprehensive account of tables in the theory of numbers, including virtually every table published from 1918 to 1941. … ►No sequel to Lehmer (1941) exists to date, but many tables of functions of number theory are included in Unpublished Mathematical Tables (1944).

33: 24.4 Basic Properties

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§24.4(iv) Finite Expansions

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24.4.15 $$B_{2n}=\frac{2n}{2^{2n}(2^{2n}-1)}\sum _{k=0}^{n-1}\left(\genfrac{}{}{0pt}{}{2n-1}{2k}\right)E_{2k},$$

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Symbols:

$B_n$: Bernoulli numbers, $E_n$: Euler numbers, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $k$: integer and $n$: integer

Permalink:

http://dlmf.nist.gov/24.4.E15

Encodings:

pMML, png, TeX

See also:

Annotations for §24.4(iv), §24.4 and Ch.24

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24.4.16 $$E_{2n}=\frac{1}{2n+1}-\sum _{k=1}^n\left(\genfrac{}{}{0pt}{}{2n}{2k-1}\right)\frac{2^{2k}(2^{2k-1}-1)B_{2k}}{k},$$

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Symbols:

$B_n$: Bernoulli numbers, $E_n$: Euler numbers, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $k$: integer and $n$: integer

Permalink:

http://dlmf.nist.gov/24.4.E16

Encodings:

pMML, png, TeX

See also:

Annotations for §24.4(iv), §24.4 and Ch.24

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§24.4(ix) Relations to Other Functions

►For the relation of Bernoulli numbers to the Riemann zeta function see §25.6, and to the Eulerian numbers see (26.14.11).

34: 24.2 Definitions and Generating Functions

§24.2 Definitions and Generating Functions

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§24.2(i) Bernoulli Numbers and Polynomials

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§24.2(ii) Euler Numbers and Polynomials

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Table 24.2.1: Bernoulli and Euler numbers.

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$n$	$B_n$	$E_n$
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35: 27.14 Unrestricted Partitions

… ►Euler’s pentagonal number theorem states that …where the exponents $1$, $2$, $5$, $7$, $12$, $15$, $\mathrm{\dots }$ are the pentagonal numbers, defined by … ►For example, the Ramanujan identity … ►

§27.14(vi) Ramanujan’s Tau Function

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36: How to Cite

… ►If you mention a specific equation (or chapter, section, …), you’ll help your readers find it by using our Permalinks & Reference numbers. …produces pdf with the equation and table numbers linking directly into the DLMF website: … ►

Permalinks & Reference numbers

… ►The following table outlines the correspondence between reference numbers as they appear in the Handbook, and the URL’s that find the same item online. … ► ►►

Item Type	Ref. Number	Permalink
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37: 27.15 Chinese Remainder Theorem

§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. …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 $(modm)$, which is correct to 20 digits. …

38: 27 Functions of Number Theory

Chapter 27 Functions of Number Theory

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39: Karl Dilcher

… ►Dilcher’s research interests include classical analysis, special functions, and elementary, combinatorial, and computational number theory. Over the years he authored or coauthored numerous papers on Bernoulli numbers and related topics, and he maintains a large on-line bibliography on the subject. …

40: 27.8 Dirichlet Characters

§27.8 Dirichlet Characters

… ►An example is the principal character (mod $k$): … ►If $\left(n,k\right)=1$, then the characters satisfy the orthogonality relation … ►A divisor $d$ of $k$ is called an induced modulus for $\chi$ if … ►If $k$ is odd, then the real characters (mod $k$) are the principal character and the quadratic characters described in the next section.