Radical_of_an

Radical of an integer

In mathematics, the radical of a positive integer n is defined as the product of the prime numbers dividing n:

$\displaystyle\mbox{rad}(n)=\prod_{\scriptstyle p\mid n\atop p~\rm prime}p.\,$

For example,

$504=2^3\cdot3^2\cdot7 \mbox{ and } \mbox{rad}(504)=2\cdot3\cdot7=42.\,$

The radical of any integer n is the largest square-free divisor of n.

Radical numbers for the first few positive integers are 1, 2, 3, 2, 5, 6, 7, 2, 3, 10, ... (sequence A007947 in OEIS).

The function $rad$ is multiplicative.

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,

$c < K_\varepsilon\, \operatorname{rad}(abc)^{1+\varepsilon}.$

References

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

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