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:

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

where $\operatorname {rad}(m)$ denotes the radical of the 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:

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

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

• Browkin, Jerzy; Brzeziński, Juliusz (1994). "Some remarks on the abc-conjecture". Math. Comp. 62 (206): 931–939. doi:10.2307/2153551. JSTOR 2153551.
• Vojta, Paul (1998). "A more general abc conjecture". arXiv:math/9806171.