Radical of an integer
In
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, ... (sequence A007947 in the OEIS).
For example,
and therefore
Properties
The function 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 -free divisor of , , which are multiplicative functions which act on prime powers as
The cases and 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
For any integer , the nilpotent elements of the finite ring are all of the multiples of .
The Dirichlet series is
References
- ^ a b Gowers, Timothy (2008). "V.1 The ABC Conjecture". The Princeton Companion to Mathematics. Princeton University Press. p. 681.
- ^ Sloane, N. J. A. (ed.). "Sequence A007947". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
- MR 1322733.