Definition:Radical of Integer
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Definition
Let $n \in \Z$ be an integer.
Definition 1
The radical of $n$ is the product of the individual prime factors of $n$.
Definition 2
The radical of $n$ is the largest square-free integer which divides $n$.
Sequence
The sequence of radicals of the integers beings:
$n$ | Decomposition | $\map \Rad n$ |
---|---|---|
$1$ | $1$ | $1$ |
$2$ | $2$ | $2$ |
$3$ | $3$ | $3$ |
$4$ | $2^2$ | $2$ |
$5$ | $5$ | $5$ |
$6$ | $2 \times 3$ | $6$ |
$7$ | $7$ | $7$ |
$8$ | $2^3$ | $2$ |
$9$ | $3^2$ | $3$ |
$10$ | $2 \times 5$ | $10$ |
$11$ | $11$ | $11$ |
$12$ | $2^2 \times 3$ | $6$ |
$13$ | $13$ | $13$ |
$14$ | $2 \times 7$ | $14$ |
$15$ | $3 \times 5$ | $15$ |
$16$ | $2^4$ | $2$ |
Also known as
The radical of an integer $n$ is also known as the square-free kernel of $n$.
Also see
- Results about radicals of integers can be found here.
Sources
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- 2008: Timothy Gowers: The Princeton Companion to Mathematics page 681, V.1 The ABC Conjecture