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##### 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: 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 $n$. …
##### 3: 25.17 Physical Applications
See Armitage (1989), Berry and Keating (1998, 1999), Keating (1993, 1999), and Sarnak (1999). …
##### 4: 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. …
##### 5: 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. …
##### 6: 27.8 Dirichlet Characters
27.8.6 $\sum_{r=1}^{\phi\left(k\right)}\chi_{r}\left(m\right)\overline{\chi}_{r}(n)=% \begin{cases}\phi\left(k\right),&m\equiv n\pmod{k},\\ 0,&\mbox{otherwise}.\end{cases}$
A Dirichlet character $\chi\pmod{k}$ is called primitive (mod $k$) if for every proper divisor $d$ of $k$ (that is, a divisor $d), there exists an integer $a\equiv 1\pmod{d}$, with $\left(a,k\right)=1$ and $\chi\left(a\right)\neq 1$. …
27.8.7 $\chi\left(a\right)=1\text{ for all a\equiv 1 (mod d)},$ $\left(a,k\right)=1$.
##### 8: 27.11 Asymptotic Formulas: Partial Sums
27.11.9 $\sum_{\begin{subarray}{c}p\leq x\\ p\equiv h\!\!\!\!\!\pmod{k}\end{subarray}}\frac{1}{p}=\frac{1}{\phi\left(k% \right)}\ln\ln x+B+O\left(\frac{1}{\ln x}\right),$
27.11.11 $\sum_{\begin{subarray}{c}p\leq x\\ p\equiv h\!\!\!\!\!\pmod{k}\end{subarray}}\frac{\ln p}{p}=\frac{1}{\phi\left(k% \right)}\ln x+O\left(1\right),$
Letting $x\to\infty$ in (27.11.9) or in (27.11.11) we see that there are infinitely many primes $p\equiv h\pmod{k}$ 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 $p\leq x$ with $p\equiv h\pmod{k}$ is asymptotic to $x/(\phi\left(k\right)\ln x)$ as $x\to\infty$.
##### 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 $x^{2}\equiv y^{2}\pmod{n}$. …
##### 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 $\ell=\ell(j,k)$ such that $j=\sigma^{\ell}(k)$, where $\sigma^{1}=\sigma$ and $\sigma^{\ell}$ is the composition of $\sigma$ with $\sigma^{\ell-1}$. …