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.

Given ${\displaystyle n\geq 3}$, let ${\displaystyle a_{1},a_{2},...,a_{n}\in \mathbb {Z} }$ satisfy three conditions:

(i) ${\displaystyle \gcd(a_{1},a_{2},...,a_{n})=1}$

(ii) ${\displaystyle a_{1}+a_{2}+...+a_{n}=0}$

(iii) no proper subsum of ${\displaystyle a_{1},a_{2},...,a_{n}}$ equals ${\displaystyle 0}$

First formulation

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

${\displaystyle \operatorname {max} (|a_{1}|,|a_{2}|,...,|a_{n}|)<C_{n,\varepsilon }\operatorname {rad} (|a_{1}|\cdot |a_{2}|\cdot \ldots \cdot |a_{n}|)^{2n-5+\varepsilon }}$

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

Second formulation

Define the quality of ${\displaystyle a_{1},a_{2},...,a_{n}}$ as

${\displaystyle q(a_{1},a_{2},...,a_{n})={\frac {\log(\operatorname {max} (|a_{1}|,|a_{2}|,...,|a_{n}|))}{\log(\operatorname {rad} (|a_{1}|\cdot |a_{2}|\cdot ...\cdot |a_{n}|))}}}$

The n conjecture states that ${\displaystyle \limsup q(a_{1},a_{2},...,a_{n})=2n-5}$.

Vojta (1998) proposed a stronger variant of the n conjecture, where setwise coprimeness of ${\displaystyle a_{1},a_{2},...,a_{n}}$ is replaced by pairwise coprimeness of ${\displaystyle a_{1},a_{2},...,a_{n}}$.

There are two different formulations of this strong n conjecture.

Given ${\displaystyle n\geq 3}$, let ${\displaystyle a_{1},a_{2},...,a_{n}\in \mathbb {Z} }$ satisfy three conditions:

(i) ${\displaystyle a_{1},a_{2},...,a_{n}}$ are pairwise coprime

(ii) ${\displaystyle a_{1}+a_{2}+...+a_{n}=0}$

(iii) no proper subsum of ${\displaystyle a_{1},a_{2},...,a_{n}}$ equals ${\displaystyle 0}$

First formulation

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

${\displaystyle \operatorname {max} (|a_{1}|,|a_{2}|,...,|a_{n}|)<C_{n,\varepsilon }\operatorname {rad} (|a_{1}|\cdot |a_{2}|\cdot \ldots \cdot |a_{n}|)^{1+\varepsilon }}$

Second formulation

Define the quality of ${\displaystyle a_{1},a_{2},...,a_{n}}$ as

${\displaystyle q(a_{1},a_{2},...,a_{n})={\frac {\log(\operatorname {max} (|a_{1}|,|a_{2}|,...,|a_{n}|))}{\log(\operatorname {rad} (|a_{1}|\cdot |a_{2}|\cdot ...\cdot |a_{n}|))}}}$

The strong n conjecture states that ${\displaystyle \limsup q(a_{1},a_{2},...,a_{n})=1}$.

Hölzl, Kleine and Stephan (2025) harvtxt error: no target: CITEREFHölzl,_Kleine_and_Stephan2025 (help) have shown that for ${\displaystyle n\geq 5}$ the above limit superior is for odd ${\displaystyle n}$ at least ${\displaystyle 5/3}$ and for even ${\displaystyle n}$ is at least ${\displaystyle 5/4}$. For the cases ${\displaystyle n=3}$ (abc-conjecture) and ${\displaystyle n=4}$, 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 ${\displaystyle n\geq 3}$. For the exact status of the case ${\displaystyle n=3}$ see the article on the abc conjecture.