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1: 27.2 Functions

… ►Euclid’s Elements (Euclid (1908, Book IX, Proposition 20)) gives an elegant proof that there are infinitely many primes. … ►An equivalent form states that the

n

th prime

p_{n}

(when the primes are listed in increasing order) is asymptotic to

n\ln n

n\to\infty

: … ►

27.2.8

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

ⓘ

Symbols:: $\phi\left(\NVar{n}\right)$ : Euler’s totient, $\equiv$ : modular equivalence, $n$ : positive integer and $a$ : primitive root
A&S Ref:: 24.3.2 II.B
Referenced by:: §27.16, §27.8
Permalink:: http://dlmf.nist.gov/27.2.E8
Encodings:: pMML, png, TeX
See also:: Annotations for §27.2(i), §27.2 and Ch.27

►and if

\phi\left(n\right)

is the smallest positive integer

f

such that

a^{f}\equiv 1\pmod{n}

, then

a

is a primitive root mod

n

. … ►

Table 27.2.2: Functions related to division.

► ►►►

$n$	$\phi\left(n\right)$	$d\left(n\right)$	$\sigma\left(n\right)$	$n$	$\phi\left(n\right)$	$d\left(n\right)$	$\sigma\left(n\right)$	$n$	$\phi\left(n\right)$	$d\left(n\right)$	$\sigma\left(n\right)$	$n$	$\phi\left(n\right)$	$d\left(n\right)$	$\sigma\left(n\right)$
…
7	6	2	8	20	8	6	42	33	20	4	48	46	22	4	72
…

►

2: 26.5 Lattice Paths: Catalan Numbers

… ►(Sixty-six equivalent definitions of

C\left(n\right)

are given in Stanley (1999, pp. 219–229).) … ►

Table 26.5.1: Catalan numbers.

► ►►►

$n$	$C\left(n\right)$	$n$	$C\left(n\right)$	$n$	$C\left(n\right)$
…
6	132	13	7 42900	20	65641 20420

► … ►

§26.5(iv) Limiting Forms

…

3: 26.9 Integer Partitions: Restricted Number and Part Size

… ►

Table 26.9.1: Partitions

p_{k}\left(n\right)

► ►►►

$n$	$k$
$n$	…
8	0	1	5	10	15	18	20	21	22	22	22
…

► … ►The conjugate partition is obtained by reflecting the Ferrers graph across the main diagonal or, equivalently, by representing each integer by a column of dots. … ►Equations (26.9.2)–(26.9.3) are examples of closed forms that can be computed explicitly for any positive integer

k

. … ►equivalently, partitions into at most

k

parts either have exactly

k

parts, in which case we can subtract one from each part, or they have strictly fewer than

k

parts. … ►

§26.9(iv) Limiting Form

…

4: 2.11 Remainder Terms; Stokes Phenomenon

… ►That the change in their forms is discontinuous, even though the function being approximated is analytic, is an example of the Stokes phenomenon. … ►In the transition through

\theta=\pi

\operatorname{erfc}\left(\sqrt{\frac{1}{2}\rho}\;c(\theta)\right)

changes very rapidly, but smoothly, from one form to the other; compare the graph of its modulus in Figure 2.11.1 in the case

\rho=100

. … ►The process just used is equivalent to re-expanding the remainder term of the original asymptotic series (2.11.24) in powers of

1/(x+5)

and truncating the new series optimally. … ►For example, using double precision

d_{20}

is found to agree with (2.11.31) to 13D. … ►Their extrapolation is based on assumed forms of remainder terms that may not always be appropriate for asymptotic expansions. …