N conjecture

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, let satisfy three conditions:
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

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:
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.
have shown that for the above limit superior is for odd at least and for even is at least. For the cases and, they did not find any nontrivial lower bounds. It is also open whether there is a common constant upper bound above the limit superiors for all. For the exact status of the case see the article on the abc conjecture.