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 the product mentioned):

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

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,

504=2^{3}\cdot 3^{2}\cdot 7

and therefore

{\mathrm {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.^[1] 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 A007948 and A058035.

One of the most striking applications of the notion of radical occurs in the abc conjecture, which states that, for any ε > 0, there exists a finite K_ε 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 }}

Furthermore, it can be shown that the nilpotent elements of $\mathbb {Z} /n\mathbb {Z}$ are all of the multiples of rad(n).

References

↑ (sequence A007947 in the OEIS)

Guy, Richard K. (2004). Unsolved Problems in Number Theory. Springer-Verlag. p. 102. ISBN 0-387-20860-7.

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Radical of an integer

Examples

Properties

See also

References