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

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.