In number theory the n conjecture is a conjecture stated by as a generalization of the abc conjecture to more than three integers.
Given
{n\ge3}
{a1,a2,...,an\inZ
(i)
\gcd(a1,a2,...,an)=1
(ii)
{a1+a2+...+an=0}
(iii) no proper subsum of
{a1,a2,...,an}
{0}
First formulation
The n conjecture states that for every
{\varepsilon>0}
C
{n}
{\varepsilon}
where\operatorname{max}(|a1|,|a2|,...,|an|)<Cn,\varepsilon\operatorname{rad}(|a1| ⋅ |a2| ⋅ ... ⋅
2n-5+\varepsilon |a n|)
\operatorname{rad}(m)
{m}
{m}
Second formulation
Define the quality of
{a1,a2,...,an}
q(a1,a2,...,an)=
log(\operatorname{max | |
(|a |
1|,|a2|,...,|an|))}{log(\operatorname{rad}(|a1| ⋅ |a2| ⋅ ... ⋅ |an|))}
\limsupq(a1,a2,...,an)=2n-5
proposed a stronger variant of the n conjecture, where setwise coprimeness of
{a1,a2,...,an}
{a1,a2,...,an}
There are two different formulations of this strong n conjecture.
Given
{n\ge3}
{a1,a2,...,an\inZ
(i)
{a1,a2,...,an}
(ii)
{a1+a2+...+an=0}
(iii) no proper subsum of
{a1,a2,...,an}
{0}
First formulation
The strong n conjecture states that for every
{\varepsilon>0}
C
{n}
{\varepsilon}
\operatorname{max}(|a1|,|a2|,...,|an|)<Cn,\varepsilon\operatorname{rad}(|a1| ⋅ |a2| ⋅ ... ⋅
1+\varepsilon |a n|)
Second formulation
Define the quality of
{a1,a2,...,an}
q(a1,a2,...,an)=
log(\operatorname{max | |
(|a |
1|,|a2|,...,|an|))}{log(\operatorname{rad}(|a1| ⋅ |a2| ⋅ ... ⋅ |an|))}
\limsupq(a1,a2,...,an)=1