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: (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 A variant called strong n conjecture or Ramaekers conjecture was proposed by . This variant of the conjecture provides a sharper quality estimate for n-tuples by replacing the assumption of setwise coprimeness of with pairwise coprimeness of . There are two formulations of the 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: .
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.
References