n conjecture
In number theory the n conjecture is a conjecture stated by (Browkin Brzeziński) as a generalization of the abc conjecture to more than three integers.
Formulations
Given [math]\displaystyle{ {n \ge 3} }[/math], let [math]\displaystyle{ {a_1,a_2,...,a_n \in \mathbb{Z}} }[/math] satisfy three conditions:
- (i) [math]\displaystyle{ \gcd(a_1,a_2,...,a_n)=1 }[/math]
- (ii) [math]\displaystyle{ {a_1 + a_2 + ... + a_n=0} }[/math]
- (iii) no proper subsum of [math]\displaystyle{ {a_1,a_2,...,a_n} }[/math] equals [math]\displaystyle{ {0} }[/math]
First formulation
The n conjecture states that for every [math]\displaystyle{ {\varepsilon \gt 0} }[/math], there is a constant [math]\displaystyle{ C }[/math], depending on [math]\displaystyle{ {n} }[/math] and [math]\displaystyle{ {\varepsilon} }[/math], such that:
[math]\displaystyle{ \operatorname{max}(|a_1|,|a_2|,...,|a_n|)\lt C_{n,\varepsilon}\operatorname{rad}(|a_1| \cdot |a_2| \cdot ... \cdot |a_n|)^{2n - 5 + \varepsilon} }[/math]
where [math]\displaystyle{ \operatorname{rad}(m) }[/math] denotes the radical of the integer [math]\displaystyle{ {m} }[/math], defined as the product of the distinct prime factors of [math]\displaystyle{ {m} }[/math].
Second formulation
Define the quality of [math]\displaystyle{ {a_1,a_2,...,a_n} }[/math] as
- [math]\displaystyle{ 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|))} }[/math]
The n conjecture states that [math]\displaystyle{ \limsup q(a_1,a_2,...,a_n)= 2n-5 }[/math].
Stronger form
(Vojta 1998) proposed a stronger variant of the n conjecture, where setwise coprimeness of [math]\displaystyle{ {a_1,a_2,...,a_n} }[/math] is replaced by pairwise coprimeness of [math]\displaystyle{ {a_1,a_2,...,a_n} }[/math].
There are two different formulations of this strong n conjecture.
Given [math]\displaystyle{ {n \ge 3} }[/math], let [math]\displaystyle{ {a_1,a_2,...,a_n \in \mathbb{Z}} }[/math] satisfy three conditions:
- (i) [math]\displaystyle{ {a_1,a_2,...,a_n} }[/math] are pairwise coprime
- (ii) [math]\displaystyle{ {a_1 + a_2 + ... + a_n=0} }[/math]
- (iii) no proper subsum of [math]\displaystyle{ {a_1,a_2,...,a_n} }[/math] equals [math]\displaystyle{ {0} }[/math]
First formulation
The strong n conjecture states that for every [math]\displaystyle{ {\varepsilon \gt 0} }[/math], there is a constant [math]\displaystyle{ C }[/math], depending on [math]\displaystyle{ {n} }[/math] and [math]\displaystyle{ {\varepsilon} }[/math], such that:
[math]\displaystyle{ \operatorname{max}(|a_1|,|a_2|,...,|a_n|)\lt C_{n,\varepsilon}\operatorname{rad}(|a_1| \cdot |a_2| \cdot ... \cdot |a_n|)^{1 + \varepsilon} }[/math]
Second formulation
Define the quality of [math]\displaystyle{ {a_1,a_2,...,a_n} }[/math] as
- [math]\displaystyle{ 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|))} }[/math]
The strong n conjecture states that [math]\displaystyle{ \limsup q(a_1,a_2,...,a_n)= 1 }[/math].
References
- Browkin, Jerzy; Brzeziński, Juliusz (1994). "Some remarks on the abc-conjecture". Math. Comp. 62 (206): 931–939. doi:10.2307/2153551. Bibcode: 1994MaCom..62..931B.
- Vojta, Paul (1998). A more general abc conjecture. Bibcode: 1998math......6171V.
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