Definition:Residue (Number Theory)

This page is about residue in the context of Number Theory. For other uses, see residue.

Definition

Let $m, n \in \N$ be natural numbers.

Let $a \in \Z$ be an integer such that $a$ is not divisible by $m$.

Then $a$ is a residue of $m$ of order $n$ if and only if:

$\exists x \in \Z: x^n \equiv a \pmod m$

where $\equiv$ denotes modulo congruence.

Nonresidue

$a$ is a nonresidue of $m$ of order $n$ if and only if there does not exist $x \in \Z$ such that:

$x^n \equiv a \pmod m$

Examples

Arbitrary Example

Consider the congruence:

$x^3 \equiv 1 \pmod 9$

This has a solution:

$x = 4$

so $1$ is a residue of $9$ of order $3$.

Also see

• Results about residues in the context of number theory can be found here.

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): residue: 2.
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): residue: 2.