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.

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