N conjecture

This article may be too technical for most readers to understand. Please help improve it to make it understandable to non-experts, without removing the technical details. (July 2015) (Learn how and when to remove this message)

This article includes a list of references, related reading, or external links, but its sources remain unclear because it lacks inline citations. Please help improve this article by introducing more precise citations. (November 2020) (Learn how and when to remove this message)

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.

• 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.{{cite journal}}: CS1 maint: multiple names: authors list (link)

• 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.