Radical of an integer


Radical of an integer

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

:displaystylembox{rad}(n)=prod_{p|n}p.,

For example,

:504=2^3cdot3^2cdot7 mbox{ and } mbox{rad}(504)=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{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 "Kε" 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

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