# even or odd

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##### 2: 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 …
##### 3: 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 $S(n)\geq 1$ 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 $n$. …Exact formulas for $r_{k}\left(n\right)$ have also been found for $k=3,5$, and $7$, and for all even $k\leq 24$. …
##### 5: 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. …
##### 6: 24.11 Asymptotic Approximations
24.11.5 $(-1)^{\left\lfloor n/2\right\rfloor-1}\frac{(2\pi)^{n}}{2(n!)}B_{n}\left(x% \right)\to\begin{cases}\cos\left(2\pi x\right),&n\text{ even},\\ \sin\left(2\pi x\right),&n\text{ odd},\end{cases}$
24.11.6 $(-1)^{\left\lfloor(n+1)/2\right\rfloor}\frac{\pi^{n+1}}{4(n!)}E_{n}\left(x% \right)\to\begin{cases}\sin\left(\pi x\right),&n\text{ even},\\ \cos\left(\pi x\right),&n\text{ odd},\end{cases}$
##### 7: 14.16 Zeros
• (c)

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

• (b)

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

• $\mathsf{Q}^{\mu}_{\nu}\left(x\right)$ 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. …
• (b)

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

• ##### 8: 24.12 Zeros
When $n(\geq 2)$ is evenWhen $n$ is odd $x^{(n)}_{1}=\frac{1}{2}$, $x^{(n)}_{2}=1$ $(n\geq 3)$, and as $n\to\infty$ with $m(\geq 1)$ fixed, … When $n(\geq 2)$ is even $y^{(n)}_{1}=1$, and as $n\to\infty$ with $m(\geq 1)$ fixed, … When $n$ is odd $y^{(n)}_{1}=\tfrac{1}{2}$, … The only polynomial $E_{n}\left(x\right)$ with multiple zeros is $E_{5}\left(x\right)=(x-\frac{1}{2})(x^{2}-x-1)^{2}$.
##### 9: 30.4 Functions of the First Kind
the sign of $\mathsf{Ps}^{m}_{n}\left(0,\gamma^{2}\right)$ being $(-1)^{(n+m)/2}$ when $n-m$ is even, and the sign of $\ifrac{\mathrm{d}\mathsf{Ps}^{m}_{n}\left(x,\gamma^{2}\right)}{\mathrm{d}x}|_{% x=0}$ being $(-1)^{(n+m-1)/2}$ when $n-m$ is odd. … with $\alpha_{k}$, $\beta_{k}$, $\gamma_{k}$ from (30.3.6), and $g_{-1}=g_{-2}=0$, $g_{k}=0$ for even $k$ if $n-m$ is odd and $g_{k}=0$ for odd $k$ if $n-m$ is even. …
##### 10: 30.16 Methods of Computation
Let $n-m$ be even. … If $n-m$ is odd, then (30.16.1) is replaced by … If $\lambda^{m}_{n}\left(\gamma^{2}\right)$ is known, then we can compute $\mathsf{Ps}^{m}_{n}\left(x,\gamma^{2}\right)$ (not normalized) by solving the differential equation (30.2.1) numerically with initial conditions $w(0)=1$, $w^{\prime}(0)=0$ if $n-m$ is even, or $w(0)=0$, $w^{\prime}(0)=1$ if $n-m$ is odd. … Let $\mathbf{A}$ be the $d\times d$ matrix given by (30.16.1) if $n-m$ is even, or by (30.16.6) if $n-m$ is odd. …