n conjecture
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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 , let satisfy three conditions:
- (i)
- (ii)
- (iii) no proper subsum of equals
First formulation
The n conjecture states that for every , there is a constant , depending on and , such that:
where denotes the radical of the integer , defined as the product of the distinct prime factors of .
Second formulation
Define the quality of as
The n conjecture states that .
Stronger form
Vojta (1998) proposed a stronger variant of the n conjecture, where setwise coprimeness of is replaced by pairwise coprimeness of .
There are two different formulations of this strong n conjecture.
Given , let satisfy three conditions:
- (i) are pairwise coprime
- (ii)
- (iii) no proper subsum of equals
First formulation
The strong n conjecture states that for every , there is a constant , depending on and , such that:
Second formulation
Define the quality of as
The strong n conjecture states that .
References
- Browkin, Jerzy; Brzeziński, Juliusz (1994). "Some remarks on the abc-conjecture". Math. Comp. 62 (206): 931–939. Bibcode:1994MaCom..62..931B. doi:10.2307/2153551. JSTOR 2153551.
- Vojta, Paul (1998). "A more general abc conjecture". International Mathematics Research Notices. 1998 (21): 1103–1116. arXiv:math/9806171. doi:10.1155/S1073792898000658. MR 1663215.
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