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1: 24.10 Arithmetic Properties

… ►where $m\equiv n\cancel{\equiv }0(modp-1)$. …valid when $m\equiv n(mod(p-1)p^{\mathrm{\ell }})$ and $n\cancel{\equiv }0(modp-1)$, where $\mathrm{\ell }(\ge 0)$ is a fixed integer. … ►

24.10.8 $$N_{2n}\equiv 0(mod{p}^{\mathrm{\ell }}),$$

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Symbols:

$\equiv$: modular equivalence, $\mathrm{\ell }$: integer, $n$: integer and $p$: prime

Referenced by:

§24.10(iv)

Permalink:

http://dlmf.nist.gov/24.10.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §24.10(iv), §24.10 and Ch.24

►valid for fixed integers $\mathrm{\ell }(\ge 1)$, and for all $n(\ge 1)$ such that $2n\cancel{\equiv }0$ $(modp-1)$ and $p^{\mathrm{\ell }}|2n$. ►

24.10.9

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Symbols:

$E_n$: Euler numbers, $\equiv$: modular equivalence, $\mathrm{\ell }$: 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

…

2: 25.17 Physical Applications

… ►See Armitage (1989), Berry and Keating (1998, 1999), Keating (1993, 1999), and Sarnak (1999). …

3: 26.21 Tables

… ►Andrews (1976) contains tables of the number of unrestricted partitions, partitions into odd parts, partitions into parts $\cancel{\equiv }\pm 2(mod5)$, partitions into parts $\cancel{\equiv }\pm 1(mod5)$, and unrestricted plane partitions up to 100. …

4: 27.16 Cryptography

… ►Thus, $y\equiv x^r(modn)$ and . … ►By the Euler–Fermat theorem (27.2.8), $x^{\varphi \left(n\right)}\equiv 1(modn)$; hence $x^{t\varphi \left(n\right)}\equiv 1(modn)$. But $y^s\equiv x^{rs}\equiv x^{1+t\varphi \left(n\right)}\equiv x(modn)$, so $y^s$ is the same as $x$ modulo $n$. …

5: 27.15 Chinese Remainder Theorem

… ►The Chinese remainder theorem states that a system of congruences $x\equiv a_1(mod{m}_1),\mathrm{\dots },x\equiv a_k(mod{m}_k)$, 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. …

6: 27.8 Dirichlet Characters

… ►

27.8.6

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Symbols:

$\chi \left(n\right)$: Dirichlet character, $\varphi \left(n\right)$: Euler’s totient, $\overline{z}$: complex conjugate, $\equiv$: modular equivalence, $k$: positive integer, $m$: positive integer and $n$: positive integer

Permalink:

http://dlmf.nist.gov/27.8.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §27.8 and Ch.27

►A Dirichlet character $\chi (modk)$ is called primitive (mod $k$) if for every proper divisor $d$ of $k$ (that is, a divisor ), there exists an integer $a\equiv 1(modd)$, with $\left(a,k\right)=1$ and $\chi \left(a\right)\ne 1$. … ►

27.8.7 $$\chi \left(a\right)=1\text{ for all }a\equiv 1\text{ (mod }d\text{)},$$ $\left(a,k\right)=1$.

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Symbols:

$\chi \left(n\right)$: Dirichlet character, $\left(m,n\right)$: greatest common divisor (gcd), $\equiv$: modular equivalence, $d$: positive integer and $k$: positive integer

Permalink:

http://dlmf.nist.gov/27.8.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §27.8 and Ch.27

…

7: 24.15 Related Sequences of Numbers

… ►

24.15.9 $$p\frac{B_n}{n}\equiv S(p-1+n,p-1)(mod{p}^2),$$ $1\le n\le p-2$,

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Symbols:

$B_n$: Bernoulli numbers, $S(n,k)$: Stirling number of the second kind, $\equiv$: modular equivalence, $n$: integer and $p$: prime

Referenced by:

§24.15(iii)

Permalink:

http://dlmf.nist.gov/24.15.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §24.15(iii), §24.15 and Ch.24

►

24.15.10 $$\frac{2n-1}{4n}{p}^2{B}_{2n}\equiv S(p+2n,p-1)(mod{p}^3),$$ $2\le 2n\le p-3$.

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Symbols:

$B_n$: Bernoulli numbers, $S(n,k)$: Stirling number of the second kind, $\equiv$: modular equivalence, $n$: integer and $p$: prime

Referenced by:

§24.15(iii)

Permalink:

http://dlmf.nist.gov/24.15.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §24.15(iii), §24.15 and Ch.24

…

8: 27.11 Asymptotic Formulas: Partial Sums

… ►

27.11.9 $$\sum _{\begin{array}{c}p\le x\\ p\equiv h(modk)\end{array}}\frac{1}{p}=\frac{1}{\varphi \left(k\right)}\ln \ln x+B+O\left(\frac{1}{\ln x}\right),$$

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Symbols:

$O\left(x\right)$: order not exceeding, $\varphi \left(n\right)$: Euler’s totient, $\equiv$: modular equivalence, $\ln z$: principal branch of logarithm function, $k$: positive integer, $p,p_1,\mathrm{\dots }$: prime numbers and $x$: real number

Referenced by:

§27.11

Permalink:

http://dlmf.nist.gov/27.11.E9

Encodings:

pMML, png, TeX

Change of Notation (effective with 1.0.10):

The notation for logarithm has been changed to $\ln$ from $\log$.

See also:

Annotations for §27.11 and Ch.27

… ►

27.11.11 $$\sum _{\begin{array}{c}p\le x\\ p\equiv h(modk)\end{array}}\frac{\ln p}{p}=\frac{1}{\varphi \left(k\right)}\ln x+O\left(1\right),$$

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $\varphi \left(n\right)$: Euler’s totient, $\equiv$: modular equivalence, $\ln z$: principal branch of logarithm function, $k$: positive integer, $p,p_1,\mathrm{\dots }$: prime numbers and $x$: real number

Referenced by:

§27.11, §27.11

Permalink:

http://dlmf.nist.gov/27.11.E11

Encodings:

pMML, png, TeX

Change of Notation (effective with 1.0.10):

The notation for logarithm has been changed to $\ln$ from $\log$.

See also:

Annotations for §27.11 and Ch.27

… ►Letting $x\to \mathrm{\infty }$ in (27.11.9) or in (27.11.11) we see that there are infinitely many primes $p\equiv h(modk)$ if $h,k$ are coprime; this is Dirichlet’s theorem on primes in arithmetic progressions. … ►Each of (27.11.13)–(27.11.15) is equivalent to the prime number theorem (27.2.3). The prime number theorem for arithmetic progressions—an extension of (27.2.3) and first proved in de la Vallée Poussin (1896a, b)—states that if $\left(h,k\right)=1$, then the number of primes $p\le x$ with $p\equiv h(modk)$ is asymptotic to $x/(\varphi \left(k\right)\ln x)$ as $x\to \mathrm{\infty }$.

9: 27.19 Methods of Computation: Factorization

… ►Type II probabilistic algorithms for factoring $n$ rely on finding a pseudo-random pair of integers $(x,y)$ that satisfy $x^2\equiv y^2(modn)$. …

10: 26.2 Basic Definitions

… ►Given a finite set $S$ with permutation $\sigma$, a cycle is an ordered equivalence class of elements of $S$ where $j$ is equivalent to $k$ if there exists an $\mathrm{\ell }=\mathrm{\ell }(j,k)$ such that $j={\sigma }^{\mathrm{\ell }}(k)$, where ${\sigma }^1=\sigma$ and ${\sigma }^{\mathrm{\ell }}$ is the composition of $\sigma$ with ${\sigma }^{\mathrm{\ell }-1}$. …