# Radical of an integer

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 this product:

The radical plays a central role in the statement of the abc conjecture.[1]

## Examples

Radical numbers for the first few 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 the OEIS).

For example,

and therefore

## Properties

The function 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*.[2] There is no known polynomial-time algorithm for computing the square-free part of an integer.[3]

The definition is generalized to the largest *t*-free divisor of *n*, , which are multiplicative functions which act on prime powers as

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

The notion of the radical occurs in the abc conjecture, which states that, for any *ε* > 0, there exists a finite *K _{ε}* such that, for all triples of coprime positive integers

*a*,

*b*, and

*c*satisfying

*a*+

*b*=

*c*,[1]

For any integer , the nilpotent elements of the finite ring are all of the multiples of .

## See also

## References

- Gowers, Timothy (2008). "V.1 The ABC Conjecture".
*The Princeton Companion to Mathematics*. Princeton University Press. p. 681. - Sloane, N. J. A. (ed.). "Sequence A007947".
*The On-Line Encyclopedia of Integer Sequences*. OEIS Foundation. - Adleman, Leonard M.; Mccurley, Kevin S. "Open Problems in Number Theoretic Complexity, II".
*Lecture Notes in Computer Science*: 9.