# Symbols:Number Theory

## Symbols used in Number Theory

### Divides, Divisor, Factor

$x \divides y$

This means:

$x$ is a divisor (or factor) of $y$

or:

$x$ divides $y$.

The $\LaTeX$ code for $x \divides y$ is x \divides y .

### Does Not Divide, Is Not a Divisor or Factor

$x \nmid y$

This means $x$ is not a divisor of $y$.

The $\LaTeX$ code for $x \nmid y$ is x \nmid y .

### Ceiling

$\ceiling x$

The ceiling function of $x$: the smallest integer greater than or equal to $x$.

The $\LaTeX$ code for $\ceiling x$ is \ceiling x .

### Floor

$\floor x$

The floor function of $x$: for $x \in \R$, the greatest integer less than or equal to $x$.

The $\LaTeX$ code for $\floor x$ is \floor x .

### Nearest Integer

$\nint x$

The nearest integer function is defined as:

$\forall x \in \R: \nint x = \begin {cases} \floor {x + \dfrac 1 2} & : x \notin 2 \Z + \dfrac 1 2 \\ x - \dfrac 1 2 & : x \in 2 \Z + \dfrac 1 2 \end{cases}$

where $\floor x$ is the floor function.

The $\LaTeX$ code for $\nint x$ is \nint x .

### Fractional Part

$\fractpart x$

Let $x \in \R$ be a real number.

Let $\floor x$ be the floor function of $x$.

The fractional part of $x$ is the difference:

$\fractpart x := x - \floor x$

The $\LaTeX$ code for $\fractpart x$ is \fractpart x .

### Coprime

$x \perp y$

This denotes the statement that $x$ is coprime to $y$.

That is:

$\gcd \set {x, y} = 1$

where $\gcd$ denotes the greatest common divisor of $x$ and $y$.

The $\LaTeX$ code for $\perp$ is \perp .

### Congruence

$\equiv$

Let $x, y \in \R$.

Then $x$ is congruent to $y$ modulo $z$ if and only if their difference is an integer multiple of $z$:

$x \equiv y \pmod z \iff \exists k \in \Z: x - y = k z$

The $\LaTeX$ code for $x \equiv y \pmod z$ is x \equiv y \pmod z .

## Deprecated Symbols

This page contains symbols which may or may not be in current use, but are either non-standard in mathematics or have been superseded by their more modern variants.

On $\mathsf{Pr} \infty \mathsf{fWiki}$ the intention is to present a consistent style, and so these symbols are to be considered deprecated.

### Divisor

$x \mid y$

This means $x$ is a divisor of $y$.

The symbol $\mid$ has been (or is in the process of being) superseded by $\divides$, which is becoming increasingly popular since many mathematicians are of the opinion that $\mid$ is overused, and hence a possible cause for confusion.

The $\LaTeX$ code for $\mid$ is \mid .

### Integer Part

$\sqbrk x$

For $x \in \R$, the greatest integer less than or equal to $x$.

That is, it is the integer part of a given real number.

$\sqbrk x$ has been (or is in the process of being) superseded by $\floor x$, due to the already widespread uses of square brackets.

The $\LaTeX$ code for $\sqbrk x$ is \sqbrk x .