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.
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
.
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
- Browkin, Jerzy; Brzeziński, Juliusz (1994). "Some remarks on the abc-conjecture". Math. Comp. 62 (206): 931–939. Bibcode:1994MaCom..62..931B. doi:10.2307/2153551. JSTOR 2153551.
- Vojta, Paul (1998). "A more general abc conjecture". International Mathematics Research Notices. 1998 (21): 1103–1116. arXiv:math/9806171. doi:10.1155/S1073792898000658. MR 1663215.
{{cite journal}}
: CS1 maint: unflagged free DOI (link)