Radical of an integer

The product of the prime factors of a given integer From Wikipedia, the free encyclopedia

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

Summarize
Perspective

The function is multiplicative (but not completely multiplicative).

The radical of any integer is the largest square-free divisor of and so also described as the square-free kernel of .[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 -free divisor of , , which are multiplicative functions which act on prime powers as

The cases and are tabulated in OEIS: A007948 and OEIS: A058035.

The notion of the radical occurs in the abc conjecture, which states that, for any , there exists a finite such that, for all triples of coprime positive integers , , and satisfying ,[1]

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

The Dirichlet series is

References

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