N conjecture explained

In number theory, the n conjecture is a conjecture stated by 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 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|} ⋅ |a _{2|} ⋅ \ldots ⋅ |a ^{2 n -5+ \varepsilon} _{n|)}

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} asq(a _{1,a} _{2,...,a} _{n)} = (log (\operatorname{max)/((|a) _{1|,|a} _{2|,...,|a} _{n|))}{log(\operatorname{rad}(|a} _{1|} ⋅ |a _{2|} ⋅ ... ⋅ |a _{n|))}} The n conjecture states that \limsup q(a _{1,a} _{2,...,a} _{n)=} 2 n -5 .

Stronger form

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 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|} ⋅ |a _{2|} ⋅ \ldots ⋅ |a ^{1+ \varepsilon} _{n|)}

Second formulation

Define the quality of

a _{1,a} _{2,...,a} _{n} asq(a _{1,a} _{2,...,a} _{n)} = (log (\operatorname{max)/((|a) _{1|,|a} _{2|,...,|a} _{n|))}{log(\operatorname{rad}(|a} _{1|} ⋅ |a _{2|} ⋅ ... ⋅ |a _{n|))}} The strong n conjecture states that \limsup q(a _{1,a} _{2,...,a} _{n)} = 1 .

References

• Jerzy Browkin . Jerzy . Browkin . Juliusz . Brzeziński . Some remarks on the abc-conjecture . Math. Comp. . 62 . 931–939 . 1994 . 10.2307/2153551 . 2153551 . 206. 1994MaCom..62..931B .
• Vojta . Paul . math/9806171 . 10.1155/S1073792898000658 . free . 21 . International Mathematics Research Notices . 1663215 . 1103–1116 . A more general conjecture . 1998. 1998 .