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.
Given n ≥ 3 , let a 1 , a 2 , . . . , a n ∈ Z satisfy three conditions:
(i)
gcd ( a 1 , a 2 , . . . , a n ) = 1 (ii)
a 1 + a 2 + . . . + a n = 0 (iii) no proper subsum of
a 1 , a 2 , . . . , a n equals
0 First formulation
The n conjecture states that for every ε > 0 , there is a constant C , depending on n and ε , such that:
max ( | a 1 | , | a 2 | , . . . , | a n | ) < C n , ε rad ( | a 1 | ⋅ | a 2 | ⋅ . . . ⋅ | a n | ) 2 n − 5 + ε
where rad ( m ) denotes the radical of the integer m , defined as the product of the distinct prime factors of m .
Second formulation
Define the quality of a 1 , a 2 , . . . , a n as
q ( a 1 , a 2 , . . . , a n ) = log ( max ( | a 1 | , | a 2 | , . . . , | a n | ) ) log ( rad ( | a 1 | ⋅ | a 2 | ⋅ . . . ⋅ | a n | ) ) The n conjecture states that lim sup q ( a 1 , a 2 , . . . , a n ) = 2 n − 5 .
Vojta (1998) proposed a stronger variant of the n conjecture, where setwise coprimeness of a 1 , a 2 , . . . , a n is replaced by pairwise coprimeness of a 1 , a 2 , . . . , a n .
There are two different formulations of this strong n conjecture.
Given n ≥ 3 , let a 1 , a 2 , . . . , a n ∈ Z satisfy three conditions:
(i)
a 1 , a 2 , . . . , a n are pairwise coprime(ii)
a 1 + a 2 + . . . + a n = 0 (iii) no proper subsum of
a 1 , a 2 , . . . , a n equals
0 First formulation
The strong n conjecture states that for every ε > 0 , there is a constant C , depending on n and ε , such that:
max ( | a 1 | , | a 2 | , . . . , | a n | ) < C n , ε rad ( | a 1 | ⋅ | a 2 | ⋅ . . . ⋅ | a n | ) 1 + ε
Second formulation
Define the quality of a 1 , a 2 , . . . , a n as
q ( a 1 , a 2 , . . . , a n ) = log ( max ( | a 1 | , | a 2 | , . . . , | a n | ) ) log ( rad ( | a 1 | ⋅ | a 2 | ⋅ . . . ⋅ | a n | ) ) The strong n conjecture states that lim sup q ( a 1 , a 2 , . . . , a n ) = 1 .
