N conjecture

In number theory the n conjecture is a conjecture stated by as a generalization of the abc conjecture to more than three integers.

Formulations
Given $$ {n \ge 3}$$, let $$ {a_1,a_2,...,a_n \in \mathbb{Z}}$$ satisfy three conditions:
 * (i) $$\gcd(a_1,a_2,...,a_n)=1$$
 * (ii) $$ {a_1 + a_2 + ... + a_n=0}$$
 * (iii) no proper subsum of $$ {a_1,a_2,...,a_n}$$ equals $$ {0}$$

First formulation

The n conjecture states that for every $$ {\varepsilon >0}$$, there is a constant $$ C $$, depending on $$ {n} $$ and $$ {\varepsilon} $$, such that: "a_1" where $$ \operatorname{rad}(m)$$ denotes the radical of the integer $$ {m} $$, defined as the product of the distinct prime factors of $$ {m} $$.

Second formulation

Define the quality of $$ {a_1,a_2,...,a_n}$$ as
 * $$ q(a_1,a_2,...,a_n)= \frac{\log(\operatorname{max}(|a_1|,|a_2|,...,|a_n|))}{\log(\operatorname{rad}(|a_1| \cdot |a_2| \cdot ... \cdot |a_n|))} $$

The n conjecture states that $$\limsup q(a_1,a_2,...,a_n)= 2n-5 $$.

Stronger form
proposed a stronger variant of the n conjecture, where setwise coprimeness of $$ {a_1,a_2,...,a_n}$$ is replaced by pairwise coprimeness of $$ {a_1,a_2,...,a_n}$$.

There are two different formulations of this strong n conjecture.

Given $$ {n \ge 3}$$, let $$ {a_1,a_2,...,a_n \in \mathbb{Z}}$$ satisfy three conditions:
 * (i) $$ {a_1,a_2,...,a_n}$$ are pairwise coprime
 * (ii) $$ {a_1 + a_2 + ... + a_n=0}$$
 * (iii) no proper subsum of $$ {a_1,a_2,...,a_n}$$ equals $$ {0}$$

First formulation

The strong n conjecture states that for every $$ {\varepsilon >0}$$, there is a constant $$ C $$, depending on $$ {n} $$ and $$ {\varepsilon} $$, such that: "a_1"

Second formulation

Define the quality of $$ {a_1,a_2,...,a_n}$$ as
 * $$ q(a_1,a_2,...,a_n)= \frac{\log(\operatorname{max}(|a_1|,|a_2|,...,|a_n|))}{\log(\operatorname{rad}(|a_1| \cdot |a_2| \cdot ... \cdot |a_n|))} $$

The strong n conjecture states that $$\limsup q(a_1,a_2,...,a_n)= 1 $$.