Radical of an integer

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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):

Examples[edit]

Radical numbers for the first few positive integers are

1, 2, 3, 2, 5, 6, 7, 2, 3, 10, ... (sequence A007947 in the OEIS).

For example,

and therefore

Properties[edit]

The function 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, , which are multiplicative functions which act on prime powers as

The cases t=3 and t=4 are tabulated in OEISA007948 and OEISA058035.

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 Kε such that, for all triples of coprime positive integers ab, and c satisfying a + b = c,

Furthermore, it can be shown that the nilpotent elements of are all of the multiples of rad(n).

See also[edit]

References[edit]

  1. ^ (sequence A007947 in the OEIS)