Radical $\operatorname{rad}(n)$ of a natural number

The radical of a natural number $n$ — also called its kernel or core — is the product of the distinct primes dividing $n$. If $n = p_1^{a_1} p_2^{a_2}\cdots p_t^{a_t}$ is the factorization of $n \ge 2$ into distinct primes, then

Equivalently, $\operatorname{rad}(n)$ is the largest squarefree divisor of $n$: it records which primes occur in $n$ while discarding all multiplicities. It is multiplicative on coprime arguments, $\operatorname{rad}(mn) = \operatorname{rad}(m)\operatorname{rad}(n)$ when $\gcd(m,n)=1$, and satisfies $\operatorname{rad}(n) \le n$ for $n \ge 1$, with equality exactly when $n$ is squarefree.

The radical is the quantity in terms of which the $abc$ conjecture of Oesterle and Masser is phrased: a triple $(a,b,c)$ of coprime positive integers with $a+b=c$ is an $abc$-hit when $\operatorname{rad}(abc) < c$, and the conjecture asserts that for each $\varepsilon>0$ only finitely many triples satisfy $\operatorname{rad}(abc)^{1+\varepsilon} < c$. Publishing the radical as a standalone definition lets every statement in that circle of ideas — the conjecture itself, its explicit form with a constant $\kappa(\varepsilon)$, the quality $\lambda(a,b,c) = \log c / \log\operatorname{rad}(abc)$, and the unconditional bounds of Stewart-Tijdeman and Stewart-Yu — refer to one and the same constant.

Formalization Note. The definition is the Finset product of Nat.primeFactors n, matching the function of the same name used in the preamble of the platform statement ABC_Conjecture and in the DeepMind formal-conjectures library. Because Nat.primeFactors 0 and Nat.primeFactors 1 are both empty, the empty-product convention gives $\operatorname{rad}(0) = \operatorname{rad}(1) = 1$; in particular $\operatorname{rad}(n) \ge 1$ for every $n$.