User:Aravind V R/sandbox1/Number/Notes

Chinese remainder theorem
Let $[a^{−1}]_{b}$ denote the multiplicative inverse of $a (mod b)$, that is, $a [a^{−1}]_{b} ≡ 1 (mod b)$. With $N = n_{1}n_{2}...n_{k}$, define $N_{j} := N⁄n_{j}$ for $j = 1, ..., k$. Then


 * $$x := \sum_{i} a_i N_i \left[\left(N_i\right)^{-1}\right]_{n_i} = a_1 N_1 \left[\left(N_1\right)^{-1}\right]_{n_1} + a_2 N_2 \left[\left(N_2\right)^{-1}\right]_{n_2} + \cdots + a_k N_k \left[\left(N_k\right)^{-1}\right]_{n_k},$$

satisfies the congrugences $x ≡ a_{i} (mod n_{i})$ for all $i = 1, ..., k$.

Fermat's little theorem
Fermat's little theorem states that if $p$ is a prime number, then for any integer $a$, the number $a^{ p} &minus; a$ is an integer multiple of $p$. In the notation of modular arithmetic, this is expressed as
 * $$a^p \equiv a \pmod p.$$

For example, if $a$ = 2 and $p$ = 7, 27 = 128, and 128 &minus; 2 = 7 × 18 is an integer multiple of 7. If $a$ is not divisible by $p$, Fermat's little theorem is equivalent to the statement that $a^{ p &minus; 1} &minus; 1$ is an integer multiple of $p$, or in symbols
 * $$a^{p-1} \equiv 1 \pmod p.$$

Carmichael number
Carmichael number is a composite number $$n$$ which satisfies the modular arithmetic congruence relation:
 * $$b^{n-1}\equiv 1\pmod{n}$$

for all integers  $$1<b<n$$ which are relatively prime to $$n$$.


 * Theorem (A. Korselt 1899): A positive composite integer $$n$$ is a Carmichael number if and only if $$n$$ is square-free, and for all prime divisors $$p$$ of $$n$$, it is true that $$p - 1 \mid n - 1$$.

Hensel's lemma
Let $$f(x)$$ be a polynomial with integer (or p-adic integer) coefficients, and let m,k be positive integers such that m ≤ k. If r is an integer such that
 * $$f(r) \equiv 0 \pmod{p^k}$$ and $$f'(r) \not\equiv 0 \pmod{p}$$

then there exists an integer s such that
 * $$f(s) \equiv 0 \pmod{p^{k+m}}$$ and $$r \equiv s \pmod{p^{k}}.$$

Furthermore, this s is unique modulo pk+m, and can be computed explicitly as
 * $$s = r + tp^k$$ where $$t = - \frac{f(r)}{p^k} \cdot (f'(r)^{-1}).$$

Euler's totient function
Euler's totient function (or Euler's phi function), denoted as $φ(n)$ or $ϕ(n)$, is an arithmetic function that counts the positive integers less than or equal to n that are relatively prime to n. The function φ(n) is multiplicative.

Euler's product formula
It states

\varphi(n) =n \prod_{p\mid n} \left(1-\frac{1}{p}\right), $$ where the product is over the distinct prime numbers dividing n.

Möbius function
For any positive integer n, define μ(n) as the sum of the primitive $n$-th roots of unity. It has values in {, , } depending on the factorization of n into prime factors:


 * μ(n) = 1 if n is a square-free positive integer with an even number of prime factors.
 * μ(n) = &minus;1 if n is a square-free positive integer with an odd number of prime factors.
 * μ(n) = 0 if n has a squared prime factor.

Von Mangoldt function
The von Mangoldt function, denoted by $Λ(n)$, is defined as


 * $$\Lambda(n) = \begin{cases} \log p & \text{if }n=p^k \text{ for some prime } p \text{ and integer } k \ge 1, \\ 0 & \text{otherwise.} \end{cases}$$

Chebyshev function
The Chebyshev function is either of two related functions. The first Chebyshev function &thetasym;(x) or &theta;(x) is given by


 * $$\vartheta(x)=\sum_{p\le x} \log p$$

with the sum extending over all prime numbers p that are less than or equal to x.

The second Chebyshev function &psi;(x) is defined similarly, with the sum extending over all prime powers not exceeding x:


 * $$ \psi(x) = \sum_{p^k\le x}\log p=\sum_{n \leq x} \Lambda(n) = \sum_{p\le x}\lfloor\log_p x\rfloor\log p, $$

where $$\Lambda$$ is the von Mangoldt function.

The second Chebyshev function can be seen to be related to the first by writing it as


 * $$\psi(x)=\sum_{p\le x} k \log p$$

where k is the unique integer such that pk ≤ x and x < pk+1.

Liouville function
The Liouville function, denoted by λ(n) and named after Joseph Liouville, is an important function in number theory.

If n is a positive integer, then λ(n) is defined as:


 * $$\lambda(n) = (-1)^{\Omega(n)},\,\! $$

where &Omega;(n) is the number of prime factors of n.

Dirichlet convolution
If f and g are two arithmetic functions (i.e. functions from the positive integers to the complex numbers), one defines a new arithmetic function f ∗ g, the Dirichlet convolution of f and g, by



\begin{align} (f*g)(n) &= \sum_{d\,\mid \,n} f(d)g\left(\frac{n}{d}\right) \\ &= \sum_{ab\,=\,n}f(a)g(b) \end{align} $$

where the sum extends over all positive divisors d of n, or equivalently over all distinct pairs (a, b) of positive integers whose product is n.

Dirichlet inverse
Inverse exists if $$f(1) \neq 0$$

Möbius inversion formula
The classic version states that if g and f are arithmetic functions satisfying
 * $$g(n)=\sum_{d\,\mid \,n}f(d)\quad\text{for every integer }n\ge 1$$

then
 * $$f(n)=\sum_{d\,\mid\, n}\mu(d)g(n/d)\quad\text{for every integer }n\ge 1$$

where μ is the Möbius function

In the language of Dirichlet convolutions, the first formula may be written as
 * $$g=f*1$$

where * denotes the Dirichlet convolution, and 1 is the constant function $$1(n)=1$$. The second formula is then written as
 * $$f=\mu * g.$$

Quadratic residue
An integer q is called a quadratic residue modulo n if it is congruent to a perfect square modulo n; i.e., if there exists an integer x such that:


 * $$x^2\equiv q \pmod{n}.$$

Otherwise, q is called a quadratic nonresidue modulo n.

Euler's criterion
Euler's criterion is a formula for determining whether an integer is a quadratic residue modulo a prime. Precisely, Let p be an odd prime and a an integer coprime to p. Then

a^{\tfrac{p-1}{2}} \equiv \begin{cases} \;\;\,1\pmod{p}& \text{ if there is an integer }x \text{ such that }a\equiv x^2 \pmod{p}\\ -1\pmod{p}& \text{ if there is no such integer.} \end{cases} $$

Euler's criterion can be concisely reformulated using the Legendre symbol:

\left(\frac{a}{p}\right) \equiv a^{(p-1)/2} \pmod p. $$

Legendre symbol
The Legendre symbol is a function of a and p defined as


 * $$\left(\frac{a}{p}\right) =

\begin{cases} 1 & \text{ if } a \text{ is a quadratic residue modulo } p \text{ and } a \not\equiv 0\pmod{p}, \\ -1 & \text{ if } a \text{ is a quadratic non-residue modulo } p, \\ 0 & \text{ if } a \equiv 0 \pmod{p}. \end{cases}$$

Legendre's original definition was by means of the explicit formula


 * $$ \left(\frac{a}{p}\right) \equiv a^{\frac{p-1}{2}} \pmod{p} \quad \text{ and } \quad\left(\frac{a}{p}\right) \in \{-1,0,1\}. $$

By Euler's criterion, which had been discovered earlier and was known to Legendre, these two definitions are equivalent.

Other
Dirichlet's theorem on arithmetic progressions