In number theory, the n conjecture is a conjecture stated by as a generalization of the abc conjecture to more than three integers.
Given , let satisfy three conditions:
First formulation
The n conjecture states that for every , there is a constant depending on and , such that: <blockquote> </blockquote> 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 .
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: <blockquote> </blockquote>
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 (abc-conjecture) 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.