# identity

(0.001 seconds)

## 1—10 of 133 matching pages

##### 1: 24.10 Arithmetic Properties
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 $E_{2n}\equiv\begin{cases}0\pmod{p^{\ell}}&\text{if }p\equiv 1\pmod{4},\\ 2\pmod{p^{\ell}}&\text{if }p\equiv 3\pmod{4},\end{cases}$
##### 2: 36.9 Integral Identities
###### §36.9 Integral Identities
36.9.9 $\left|\Psi^{(\mathrm{E})}\left(x,y,z\right)\right|^{2}=\frac{8\pi^{2}}{3^{2/3}% }\int_{0}^{\infty}\int_{0}^{2\pi}\Re\left(\mathrm{Ai}\left(\frac{1}{3^{1/3}}% \left(x+iy+2zu\exp\left(i\theta\right)+3u^{2}\exp\left(-2i\theta\right)\right)% \right)\*\mathrm{Bi}\left(\frac{1}{3^{1/3}}\left(x-iy+2zu\exp\left(-i\theta% \right)+3u^{2}\exp\left(2i\theta\right)\right)\right)\right)u\mathrm{d}u% \mathrm{d}\theta.{}$
##### 3: 26.21 Tables
Andrews (1976) contains tables of the number of unrestricted partitions, partitions into odd parts, partitions into parts $\not\equiv\pm 2\pmod{5}$, partitions into parts $\not\equiv\pm 1\pmod{5}$, and unrestricted plane partitions up to 100. …
##### 4: 27.16 Cryptography
Thus, $y\equiv x^{r}\pmod{n}$ and $1\leq y. … 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 $n$. …
##### 6: 24.5 Recurrence Relations
###### §24.5(iii) Inversion Formulas
In each of (24.5.9) and (24.5.10) the first identity implies the second one and vice-versa. …
##### 7: 15.17 Mathematical Applications
###### §15.17(iv) Combinatorics
In combinatorics, hypergeometric identities classify single sums of products of binomial coefficients. …
##### 8: 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. …
##### 9: 17.17 Physical Applications
In exactly solved models in statistical mechanics (Baxter (1981, 1982)) the methods and identities of §17.12 play a substantial role. …
##### 10: 17.18 Methods of Computation
Lehner (1941) uses Method (2) in connection with the Rogers–Ramanujan identities. …