n conjecture

{{short description|Generalization of the abc conjecture to more than three integers}}

In number theory, the n conjecture is a conjecture stated by {{Harvtxt|Browkin|Brzeziński|1994}} as a generalization of the abc conjecture to more than three integers.

Formulations

Given $n \ge 3$ , let $a_1,a_2,...,a_n \in \mathbb{Z}$ 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

{{Harvtxt|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 $n \ge 3$ , 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 $\limsup q(a_1,a_2,...,a_n) = 1$ .

References

{{Cite journal |authorlink=Jerzy Browkin |first1=Jerzy |last1=Browkin |first2=Juliusz |last2=Brzeziński | title=Some remarks on the abc-conjecture | journal=Math. Comp. | volume=62 | pages=931–939 | year=1994 | doi=10.2307/2153551 | jstor=2153551 | issue=206|bibcode=1994MaCom..62..931B }}
{{cite journal

| last = Vojta | first = Paul

| arxiv = math/9806171

| doi = 10.1155/S1073792898000658 |doi-access=free

| issue = 21

| journal = International Mathematics Research Notices

| mr = 1663215

| pages = 1103–1116

| title = A more general {{mvar|abc}} conjecture

| year = 1998| volume = 1998

}}

Category:Conjectures

Category:Unsolved problems in number theory

Category:Abc conjecture