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 ${\displaystyle n\geq 3}$ , 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 ${\displaystyle \limsup q(a_{1},a_{2},...,a_{n})=2n-5}$ .

Stronger form

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.

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.