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 this product:

The radical plays a central role in the statement of the abc conjecture.[1]

Examples[edit]

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[edit]

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 OEISA007948 and OEISA058035.

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 coprime positive integers , , and satisfying ,[1]

For any integer , the nilpotent elements of the finite ring are all of the multiples of .

References[edit]

  1. ^ a b Gowers, Timothy (2008). "V.1 The ABC Conjecture". The Princeton Companion to Mathematics. Princeton University Press. p. 681.
  2. ^ Sloane, N. J. A. (ed.). "Sequence A007947". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
  3. ^ Adleman, Leonard M.; McCurley, Kevin S. "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. pp. 291–322. CiteSeerX 10.1.1.48.4877. doi:10.1007/3-540-58691-1_70. MR 1322733.