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 $n\geq 3$, 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:

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

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

Second formulation

Define the quality of $a_{1},a_{2},...,a_{n}$ 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 $n\geq 3$, 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:

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

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 $\limsup q(a_{1},a_{2},...,a_{n})=1$.

Hölzl, Kleine and Stephan (2025) have shown that for $n\geq 5$ 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 $n\geq 3$. For the exact status of the case $n=3$ see the article on the abc conjecture.

References

• Browkin, Jerzy; Brzeziński, Juliusz (1994). "Some remarks on the abc-conjecture". Math. Comp. 62 (206): 931–939. Bibcode:1994MaCom..62..931B. doi:10.2307/2153551. JSTOR 2153551.
• Hölzl, Rupert; Kleine, Sören; Stephan, Frank (2025). "Improved lower bounds for strong n-conjectures". Journal of the Australian Mathematical Society. 119: 61–81. arXiv:2409.13439. doi:10.1017/S1446788725000084.

• Vojta, Paul (1998). "A more general abc conjecture". International Mathematics Research Notices. 1998 (21): 1103–1116. arXiv:math/9806171. doi:10.1155/S1073792898000658. MR 1663215.