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

:$displaystylembox\left\{rad\right\}\left(n\right)=prod_\left\{p|n\right\}p.,$

For example,

:$504=2^3cdot3^2cdot7 mbox\left\{ and \right\} mbox\left\{rad\right\}\left(504\right)=2cdot3cdot7=42.,$

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 [http://www.research.att.com/~njas/sequences/A007947 A007947] in OEIS).

The function $mbox\left\{rad\right\}$ 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 "Kε" such that, for all triples of coprime positive integers "a", "b", and "c" satisfying "a" + "b" = "c",

:$c < K_varepsilon, operatorname\left\{rad\right\}\left(abc\right)^\left\{1+varepsilon\right\}.$

References

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