Radical of an integer

In number theory, the radical of an integer is the product of its unique prime factors. The radical of an integer  is written . 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  is given by

This can be read in plain language as "the product of all prime numbers  that evenly divide ".

 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, ... .

 Properties 
The radical of a number is the largest square-free factor of that number.  if and only if  is square-free.

For any two integers  and ,

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

In ring theory,  is the greatest common divisor of the nilpotent elements of the ring of integers modulo .

 Sources 

Category:Number theory