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:
{\displaystyle \displaystyle \mathrm {rad} (n)=\prod _{\scriptstyle p\mid n \atop p{\text{ prime}}}p}
The radical plays a central role in the statement of the abc conjecture.
[1] Radical numbers for the first few positive integers are For example,
rad ( 504 ) = 2 ⋅ 3 ⋅ 7 = 42
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
, which are multiplicative functions which act on prime powers as
{\displaystyle \mathrm {rad} _{t}(p^{e})=p^{\mathrm {min} (e,t-1)}}
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
{\displaystyle c , the nilpotent elements of the finite ring The Dirichlet series is