Generalization of the abc conjecture to more than three integers
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.
Formulations[edit]
Given
, let
satisfy three conditions:
- (i)
![{\displaystyle \gcd(a_{1},a_{2},...,a_{n})=1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/22f303c9c91ab28bec7e653cd510d21d60f0f38e)
- (ii)
![{\displaystyle {a_{1}+a_{2}+...+a_{n}=0}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7908cfdb1afd348966262135011eaebb9cd736a5)
- (iii) no proper subsum of
equals ![{\displaystyle {0}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8f8f8566bdc86ddf764fdd921b5f6460a28f2fb6)
First formulation
The n conjecture states that for every
, there is a constant
, depending on
and
, such that:
where
denotes the radical of the integer
, defined as the product of the distinct prime factors of
.
Second formulation
Define the quality of
as
![{\displaystyle q(a_{1},a_{2},...,a_{n})={\frac {\log(\operatorname {max} (|a_{1}|,|a_{2}|,...,|a_{n}|))}{\log(\operatorname {rad} (|a_{1}|\cdot |a_{2}|\cdot ...\cdot |a_{n}|))}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3161dea8ce253a05cc204bef5efce768af94b7c3)
The n conjecture states that
.
Stronger form[edit]
Vojta (1998) 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:
- (i)
are pairwise coprime
- (ii)
![{\displaystyle {a_{1}+a_{2}+...+a_{n}=0}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7908cfdb1afd348966262135011eaebb9cd736a5)
- (iii) no proper subsum of
equals ![{\displaystyle {0}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8f8f8566bdc86ddf764fdd921b5f6460a28f2fb6)
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
![{\displaystyle q(a_{1},a_{2},...,a_{n})={\frac {\log(\operatorname {max} (|a_{1}|,|a_{2}|,...,|a_{n}|))}{\log(\operatorname {rad} (|a_{1}|\cdot |a_{2}|\cdot ...\cdot |a_{n}|))}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3161dea8ce253a05cc204bef5efce768af94b7c3)
The strong n conjecture states that
.
References[edit]