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 $a_1,a_2,\mathrm{...},a_n\in \mathbb{Z}$ satisfy three conditions:

(i) $\gcd (a_1,a_2,\mathrm{...},a_n)=1$

(ii) $a_1+a_2+\mathrm{...}+a_n=0$

(iii) no proper subsum of $a_1,a_2,\mathrm{...},a_n$ equals $0$

First formulation

The n conjecture states that for every $\epsilon >0$, there is a constant $C$ depending on $n$ and $\epsilon$, such that:

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

where $\mathrm{rad}(m)$ denotes the radical of an integer $m$, defined as the product of the distinct prime factors of $m$.

Second formulation

Define the quality of $a_1,a_2,\mathrm{...},a_n$ as

$q(a_1,a_2,\mathrm{...},a_n)=\frac{\log (\max (|a_1|,|a_2|,\mathrm{...},|a_n|))}{\log (\mathrm{rad}(|a_1|\cdot |a_2|\cdot \mathrm{...}\cdot |a_n|))}$

The n conjecture states that $\mathrm{lim\; sup}q(a_1,a_2,\mathrm{...},a_n)=2n-5$.

Stronger form

Vojta (1998) proposed a stronger variant of the n conjecture, where setwise coprimeness of $a_1,a_2,\mathrm{...},a_n$ is replaced by pairwise coprimeness of $a_1,a_2,\mathrm{...},a_n$.

There are two different formulations of this strong n conjecture.

Given $n\ge 3$, let $a_1,a_2,\mathrm{...},a_n\in \mathbb{Z}$ satisfy three conditions:

(i) $a_1,a_2,\mathrm{...},a_n$ are pairwise coprime

(ii) $a_1+a_2+\mathrm{...}+a_n=0$

(iii) no proper subsum of $a_1,a_2,\mathrm{...},a_n$ equals $0$

First formulation

The strong n conjecture states that for every $\epsilon >0$, there is a constant $C$ depending on $n$ and $\epsilon$, such that:

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

Second formulation

Define the quality of $a_1,a_2,\mathrm{...},a_n$ as

$q(a_1,a_2,\mathrm{...},a_n)=\frac{\log (\max (|a_1|,|a_2|,\mathrm{...},|a_n|))}{\log (\mathrm{rad}(|a_1|\cdot |a_2|\cdot \mathrm{...}\cdot |a_n|))}$

The strong n conjecture states that $\mathrm{lim\; sup}q(a_1,a_2,\mathrm{...},a_n)=1$.

Hölzl, Kleine and Stephan (2025) have shown that for $n\ge 5$ the above limit superior is for odd $n$ at least $5/3$ and for even $n$ is at least $5/4$. For the cases $n=3$ (abc-conjecture) and $n=4$, they did not find any nontrivial lower bounds. It is also open whether there is a common constant upper bound above the limit superiors for all $n\ge 3$. For the exact status of the case $n=3$ see the article on the abc conjecture.