N conjecture

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 an 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.