Radical of an integer explained

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:

$$\backslash displaystyle\backslash mathrm(n)=\backslash prod\_p$$

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, ... .

For example,$$504\; =\; 2^3\; \backslash cdot\; 3^2\; \backslash cdot\; 7$$

and therefore$$\backslash operatorname(504)\; =\; 2\; \backslash cdot\; 3\; \backslash cdot\; 7\; =\; 42$$

Properties

The function

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

The definition is generalized to the largest

t -free divisor of n , rad _{t} , which are multiplicative functions which act on prime powers as

$$\backslash mathrm\_t(p^e)\; =\; p^$$

The cases

t =3 and t =4 are tabulated in and .

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 ,^[1]

For any integer

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

The Dirichlet series is

\prod _{p} \left( 1+ (p ^{1-s})/(1- p ^{-s}) \right) = \sum ^{inf ty} _{n =1} (\operatorname{rad)/((n)}{n) ^{s}}

Notes and References

1. Book: Gowers, Timothy . V.1 The ABC Conjecture . The Princeton Companion to Mathematics . 681 . Timothy Gowers . Princeton University Press . 2008. https://books.google.com/books?id=ZOfUsvemJDMC&pg=PA681. The Princeton Companion to Mathematics .
2. Book: Adleman, Leonard M.. Leonard Adleman . McCurley. Kevin S.. Kevin McCurley (cryptographer). Open Problems in Number Theoretic Complexity, II. Algorithmic Number Theory: First International Symposium, ANTS-I Ithaca, NY, USA, May 6–9, 1994, Proceedings. Lecture Notes in Computer Science. 877. Springer. 1322733. 291–322. 10.1007/3-540-58691-1_70. 10.1.1.48.4877.