radical of an integer

{{Short description|The product of the prime factors of a given 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:

$\displaystyle\mathrm{rad}(n)=\prod_{\scriptstyle p\mid n\atop p\text{ prime}}p$

The radical plays a central role in the statement of the abc conjecture.{{cite book |contribution=V.1 The ABC Conjecture |title=The Princeton Companion to Mathematics |page=681 |first=Timothy |last=Gowers|author-link=Timothy Gowers |publisher=Princeton University Press |year=2008|contribution-url=https://books.google.com/books?id=ZOfUsvemJDMC&pg=PA681|title-link=The Princeton Companion to Mathematics }}

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, ... {{OEIS|id=A007947}}.

For example,

$504 = 2^3 \cdot 3^2 \cdot 7$

and therefore

$\operatorname{rad}(504) = 2 \cdot 3 \cdot 7 = 42$

Properties

The function $\mathrm{rad}$ 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 $n$ .{{cite OEIS|A007947}} There is no known polynomial-time algorithm for computing the square-free part of an integer.{{Cite book|last1=Adleman|first1=Leonard M.|author1-link= Leonard Adleman |last2=McCurley|first2=Kevin S.|author2-link=Kevin McCurley (cryptographer)|contribution=Open Problems in Number Theoretic Complexity, II|title=Algorithmic Number Theory: First International Symposium, ANTS-I Ithaca, NY, USA, May 6–9, 1994, Proceedings|series=Lecture Notes in Computer Science|date=1994 |volume=877|publisher=Springer|mr=1322733|pages=291–322|doi=10.1007/3-540-58691-1_70|isbn=978-3-540-58691-3 |citeseerx=10.1.1.48.4877}}

The definition is generalized to the largest $t$ -free divisor of $n$ , $\mathrm{rad}_t$ , which are multiplicative functions which act on prime powers as

$\mathrm{rad}_t(p^e) = p^{\mathrm{min}(e, t - 1)}$

The cases $t=3$ and $t=4$ are tabulated in {{OEIS2C|A007948}} and {{OEIS2C|A058035}}.

The notion of the radical occurs in the abc conjecture, which states that, for any $\varepsilon > 0$ , there exists a finite $K_\varepsilon$ such that, for all triples of coprime positive integers $a$ , $b$ , and $c$ satisfying $a+b=c$ ,

$c < K_\varepsilon\, \operatorname{rad}(abc)^{1 + \varepsilon}$

For any integer $n$ , the nilpotent elements of the finite ring $\mathbb{Z}/n\mathbb{Z}$ are all of the multiples of $\operatorname{rad}(n)$ .

The Dirichlet series is

: $\prod_p \left(1+\frac{p^{1-s}}{1-p^{-s}}\right) = \sum_{n=1}^{\infty} \frac{\operatorname{rad}(n)}{n^s}$

References

Category:Multiplicative functions

Category:Abc conjecture

de:Zahlentheoretische Funktion#Multiplikative Funktionen