Harman Patil (Editor)

Radical of an integer

Updated on
Edit
Like
Comment
Share on FacebookTweet on TwitterShare on LinkedInShare on Reddit

In number theory, the radical of a positive integer n is defined as the product of the distinct prime numbers dividing n (each prime factor of n occurs exactly once as a factor of the product mentioned):

Contents

r a d ( n ) = p n p  prime p

Examples

Radical numbers for the first few positive integers are

1, 2, 3, 2, 5, 6, 7, 2, 3, 10, ... (sequence A007947 in the OEIS).

For example,

504 = 2 3 3 2 7

and therefore

r a d ( 504 ) = 2 3 7 = 42

Properties

The function r a d is multiplicative (but not completely multiplicative).

The radical of any integer n is the largest square-free divisor of n and so also described as the square-free kernel of n. The definition is generalized to the largest t-free divisor of n, r a d t , which are multiplicative functions which act on prime powers as

r a d t ( p e ) = p m i n ( e , t 1 )

The cases t=3 and t=4 are tabulated in  A007948 and  A058035.

One of the most striking applications of the notion of radical occurs in the abc conjecture, which states that, for any ε > 0, there exists a finite Kε such that, for all triples of coprime positive integers ab, and c satisfying a + b = c,

c < K ε rad ( a b c ) 1 + ε

Furthermore, it can be shown that the nilpotent elements of Z / n Z are all of the multiples of rad(n).

References

Radical of an integer Wikipedia


Similar Topics