Radical of an integer

This page is about the function in number theory. For the operation written with the radical symbol, see square root.

In number theory, the radical of an integer is the product of its unique prime factors. The radical of an integer [math]\displaystyle{ n }[/math] is written [math]\displaystyle{ \mathrm{rad}\, n }[/math]. The radical is an important part of the abc conjecture, one of the most important unsolved problems in mathematics.

Definition

In mathematical notation, the radical of an integer [math]\displaystyle{ x }[/math] is given by [math]\displaystyle{ \mathrm{rad}\, n = \prod_{p\mid n\in \mathbb{P}} p }[/math] This can be read in plain language as "the product of all prime numbers [math]\displaystyle{ p }[/math] that evenly divide [math]\displaystyle{ n }[/math]".

Examples

The radicals of the first positive integers are

1, 2, 3, 2, 5, 6, 7, 2, 3, 10, 11, 6, 13, 14, 15, 2, 17, 6, 19, 10, 21, 22, 23, 6, 5, 26, 3, 14, 29, 30, 31, 2, 33, 34, 35, 6, 37, 38, 39, 10, 41, 42, 43, 22, 15, 46, 47, 6, 7, 10, ... (sequence A007947 in OEIS).

Properties

The radical of a number is the largest square-free factor of that number. [math]\displaystyle{ n = \mathrm{rad}\, n }[/math] if and only if [math]\displaystyle{ n }[/math] is square-free.

For any two integers [math]\displaystyle{ a }[/math] and [math]\displaystyle{ b }[/math],

[math]\displaystyle{ \mathrm{rad}\, ab = \frac{(\mathrm{rad}\, a)(\mathrm{rad}\, b)}{\mathrm{rad}\,(\gcd(a,b))} }[/math]

It follows from this that the radical is an incompletely multiplicative function.

In ring theory, [math]\displaystyle{ \mathrm{rad}\, n }[/math] is the greatest common divisor of the nilpotent elements of the ring of integers modulo [math]\displaystyle{ n }[/math].