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1: 27.9 Quadratic Characters

… ►For an odd prime $p$, the Legendre symbol $(n|p)$ is defined as follows. … ►If $p,q$ are distinct odd primes, then the quadratic reciprocity law states that ►

27.9.3 $$(p|q)(q|p)={(-1)}^{(p-1)(q-1)/4}.$$

ⓘ

Symbols:

$(n|p)$: Legendre symbol, $p$: odd prime and $q$: odd prime

Referenced by:

§27.9

Permalink:

http://dlmf.nist.gov/27.9.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §27.9 and Ch.27

►If an odd integer $P$ has prime factorization $P={\prod }_{r=1}^{\nu \left(n\right)}{p}_r^{a_r}$, then the Jacobi symbol $(n|P)$ is defined by $(n|P)={\prod }_{r=1}^{\nu \left(n\right)}{(n|p_r)}^{a_r}$, with $(n|1)=1$. …Both (27.9.1) and (27.9.2) are valid with $p$ replaced by $P$; the reciprocity law (27.9.3) holds if $p,q$ are replaced by any two relatively prime odd integers $P,Q$.

2: 27.13 Functions

… ►Every even integer $n>4$ is the sum of two odd primes. In this case, $S(n)$ is the number of solutions of the equation $n=p+q$, where $p$ and $q$ are odd primes. Goldbach’s assertion is that $S(n)\ge 1$ for all even $n>4$. …Vinogradov (1937) proves that every sufficiently large odd integer is the sum of three odd primes, and Chen (1966) shows that every sufficiently large even integer is the sum of a prime and a number with no more than two prime factors. … ►By similar methods Jacobi proved that $r_4\left(n\right)=8{\sigma }_1\left(n\right)$ if $n$ is odd, whereas, if $n$ is even, $r_4\left(n\right)=24$ times the sum of the odd divisors of $n$. …

3: 14.27 Zeros

… ►

$\mu ,\nu \in \mathbb{Z}$, , and $\nu$ is odd.

…

4: 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. …

5: 10.59 Integrals

…

6: 23.18 Modular Transformations

… ►Here e and o are generic symbols for even and odd integers, respectively. In particular, if $a-1,b,c$, and $d-1$ are all even, then …

7: 29.12 Definitions

… ►

Table 29.12.1: Lamé polynomials.

► ►►►►►►►

$\nu$	eigenvalue $h$	eigenfunction $w(z)$	polynomial form	real period	imag. period	parity of $w(z)$	parity of $w(z-K)$	parity of $w(z-K-\mathrm{i}{K}^{\prime })$
…
$2n+1$	$a_{\nu }^{2m+1}\left(k^2\right)$	${\mathrm{𝑠𝐸}}_{\nu }^m(z,k^2)$	$\mathrm{sn}P({\mathrm{sn}}^2)$	$4K$	$2\mathrm{i}{K}^{\prime }$	odd	even	even
…
$2n+2$	$b_{\nu }^{2m+2}\left(k^2\right)$	${\mathrm{𝑠𝑐𝐸}}_{\nu }^m(z,k^2)$	$\mathrm{sn}\mathrm{cn}P({\mathrm{sn}}^2)$	$2K$	$4\mathrm{i}{K}^{\prime }$	odd	odd	even
$2n+2$	$a_{\nu }^{2m+1}\left(k^2\right)$	${\mathrm{𝑠𝑑𝐸}}_{\nu }^m(z,k^2)$	$\mathrm{sn}\mathrm{dn}P({\mathrm{sn}}^2)$	$4K$	$4\mathrm{i}{K}^{\prime }$	odd	even	odd
$2n+2$	$b_{\nu }^{2m+1}\left(k^2\right)$	${\mathrm{𝑐𝑑𝐸}}_{\nu }^m(z,k^2)$	$\mathrm{cn}\mathrm{dn}P({\mathrm{sn}}^2)$	$4K$	$2\mathrm{i}{K}^{\prime }$	even	odd	odd
$2n+3$	$b_{\nu }^{2m+2}\left(k^2\right)$	${\mathrm{𝑠𝑐𝑑𝐸}}_{\nu }^m(z,k^2)$	$\mathrm{sn}\mathrm{cn}\mathrm{dn}P({\mathrm{sn}}^2)$	$2K$	$2\mathrm{i}{K}^{\prime }$	odd	odd	odd

►

…

8: 26.13 Permutations: Cycle Notation

… ►For the example (26.13.2), this decomposition is given by $\left(1,3,2,5,7\right)\left(6,8\right)=\left(1,3\right)\left(2,3\right)\left(2,5\right)\left(5,7\right)\left(6,8\right).$ ►A permutation is even or odd according to the parity of the number of transpositions. The sign of a permutation is $+$ if the permutation is even, $-$ if it is odd. …

9: 14.16 Zeros

… ►

$\mu >0$, , and $m-n$ is odd.

… ► ${𝖰}_{\nu }^{\mu }\left(x\right)$ has $\max (\lceil \nu -|\mu |\rceil ,0)+k$ zeros in the interval $(-1,1)$, where $k$ can take one of the values $-1$, $0$, $1$, $2$, subject to $\max (\lceil \nu -|\mu |\rceil ,0)+k$ being even or odd according as $\cos \left(\nu \pi \right)$ and $\cos \left(\mu \pi \right)$ have opposite signs or the same sign. … ►

$\mu \le \nu$, $\mu \notin \mathbb{Z}$, and $\lfloor \mu \rfloor$ is odd.

…

10: 24.12 Zeros

… ►When $n$ is odd $x_1^{(n)}=\frac{1}{2}$, $x_2^{(n)}=1$ $(n\ge 3)$, and as $n\to \mathrm{\infty }$ with $m(\ge 1)$ fixed, … ►When $n$ is odd $y_1^{(n)}=\frac{1}{2}$, … ►The only polynomial $E_n\left(x\right)$ with multiple zeros is $E_5\left(x\right)=(x-\frac{1}{2}){(x^2-x-1)}^2$.