Wikipedia / Radical of an integer
Wikipedia / 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.
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, ... .
Properties
is multiplicative (but not completely multiplicative).
is the largest square-free and so also described as the square-free kernel. There is no known polynomial-time algorithm for computing the square-free part of an integer.
, which are multiplicative functions which act on prime powers as
are tabulated in and .
The notion of the radical occurs in the abc conjecture such that, for all triples of coprime,
, the nilpotent elements of the finite ring.
The Dirichlet series is
Notation and name
The notation for the function was likely introduced by Joseph Oesterlé, in the same paper where the abc conjecture was published. The name is related to the radical of an ideal.