# 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`

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### 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`

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### 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`

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### 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`

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### 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`

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### 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`

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### 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`

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page contains symbols which may or may not be in current use, but are either non-standard in mathematics or have been superseded by their more modern variants.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

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### 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`

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