quintuple product identity

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1: 17.8 Special Cases of ${{}_{r}\psi_{r}}$ Functions

… ►

Jacobi’s Triple Product

… ►

Quintuple Product Identity

…

2: 24.10 Arithmetic Properties

… ►The denominator of

B_{2n}

is the product of all these primes

p

. … ►where

m\equiv n\not\equiv 0\pmod{p-1}

. …valid when

m\equiv n\pmod{(p-1)p^{\ell}}

and

n\not\equiv 0\pmod{p-1}

, where

\ell(\geq 0)

is a fixed integer. … ►valid for fixed integers

\ell(\geq 1)

, and for all

n(\geq 1)

such that

2n\not\equiv 0

\pmod{p-1}

and

p^{\ell}\mathbin{|}2n

. ►

24.10.9

E_{2n}\equiv\begin{cases}0\pmod{p^{\ell}}&\text{if }p\equiv 1\pmod{4},\\ 2\pmod{p^{\ell}}&\text{if }p\equiv 3\pmod{4},\end{cases}

ⓘ

Symbols:: $E_{\NVar{n}}$ : Euler numbers, $\equiv$ : modular equivalence, $\ell$ : integer, $n$ : integer and $p$ : prime
Referenced by:: §24.10(iv)
Permalink:: http://dlmf.nist.gov/24.10.E9
Encodings:: pMML, png, TeX
See also:: Annotations for §24.10(iv), §24.10 and Ch.24

…

3: 27.15 Chinese Remainder Theorem

… ►The Chinese remainder theorem states that a system of congruences

x\equiv a_{1}\pmod{m_{1}},\dots,x\equiv a_{k}\pmod{m_{k}}

, always has a solution if the moduli are relatively prime in pairs; the solution is unique (mod

m

), where

m

is the product of the moduli. … ►Their product

m

has 20 digits, twice the number of digits in the data. …

4: 1.1 Special Notation

… ► ►►►►

$x, y$	real variables.
…
$\left\langle f,g\right\rangle$	inner, or scalar, product for real or complex vectors or functions.
…
$\mathbf{I}$	identity matrix
…

…

5: 1.2 Elementary Algebra

… ►

§1.2(v) Matrices, Vectors, Scalar Products, and Norms

… ►The transpose of the product is … ►Column vectors

\mathbf{u}

and

\mathbf{v}

of the same length

n

have a scalar product … ►The scalar product has properties … ►The identity matrix

\mathbf{I}

, is defined as …

6: 27.16 Cryptography

… ►The primes are kept secret but their product

n=pq

, an 800-digit number, is made public. … ►Thus,

y\equiv x^{r}\pmod{n}

and

1\leq y<n

. … ►By the Euler–Fermat theorem (27.2.8),

x^{\phi\left(n\right)}\equiv 1\pmod{n}

; hence

x^{t\phi\left(n\right)}\equiv 1\pmod{n}

. But

y^{s}\equiv x^{rs}\equiv x^{1+t\phi\left(n\right)}\equiv x\pmod{n}

, so

y^{s}

is the same as

x

modulo

n

. …

7: 27.4 Euler Products and Dirichlet Series

§27.4 Euler Products and Dirichlet Series

►The fundamental theorem of arithmetic is linked to analysis through the concept of the Euler product. Every multiplicative

f

satisfies the identity …In this case the infinite product on the right (extended over all primes

p

) is also absolutely convergent and is called the Euler product of the series. If

f(n)

is completely multiplicative, then each factor in the product is a geometric series and the Euler product becomes …

8: 15.17 Mathematical Applications

… ►

§15.17(iv) Combinatorics

►In combinatorics, hypergeometric identities classify single sums of products of binomial coefficients. …

9: 24.19 Methods of Computation

… ►Another method is based on the identities ►

24.19.1

N_{2n}=\frac{2(2n)!}{(2\pi)^{2n}}\left(\prod_{p-1\mathbin{|}2n}p\right)\left(% \prod_{p}\frac{p^{2n}}{p^{2n}-1}\right),

ⓘ

Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $!$ : factorial (as in $n!$ ), $n$ : integer and $p$ : prime
Referenced by:: §24.19(i)
Permalink:: http://dlmf.nist.gov/24.19.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §24.19(i), §24.19 and Ch.24

►

D_{2n}=\prod_{p-1\mathbin{|}2n}p,

… ►If

\widetilde{N}_{2n}

denotes the right-hand side of (24.19.1) but with the second product taken only for

p\leq\left\lfloor(\pi e)^{-1}2n\right\rfloor+1

, then

N_{2n}=\left\lceil\widetilde{N}_{2n}\right\rceil

for

n\geq 2

. … ►For number-theoretic applications it is important to compute

B_{2n}\pmod{p}

for

2n\leq p-3

; in particular to find the irregular pairs

(2n,p)

for which

B_{2n}\equiv 0\pmod{p}

. …

10: 21.6 Products

§21.6 Products

►

§21.6(i) Riemann Identity

… ►Then …This is the Riemann identity. …Many identities involving products of theta functions can be established using these formulas. …