n conjecture

In number theory the n conjecture is a conjecture stated by Browkin & Brzeziński (1994) 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:

$\operatorname{max}(|a_1|,|a_2|,...,|a_n|)< C_{n,\varepsilon}\operatorname{rad}(|a_1| \cdot |a_2| \cdot ... \cdot |a_n|)^{2n - 5 + \varepsilon}$

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

Vojta (1998) 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:

$\operatorname{max}(|a_1|,|a_2|,...,|a_n|)< C_{n,\varepsilon}\operatorname{rad}(|a_1| \cdot |a_2| \cdot ... \cdot |a_n|)^{1 + \varepsilon}$

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 $\limsup q(a_1,a_2,...,a_n)= 1$.

References

• Browkin, Jerzy; Brzeziński, Juliusz (1994). "Some remarks on the abc-conjecture". Math. Comp. 62 (206): 931–939. doi:10.2307/2153551. JSTOR 2153551.
• Vojta, Paul (1998). "A more general abc conjecture". arXiv:math/9806171.