n conjecture

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

Given ${\displaystyle {n\geq 3}}$ , let ${\displaystyle {a_{1},a_{2},...,a_{n}\in \mathbb {Z} }}$  satisfy three conditions:

(i) ${\displaystyle \gcd(a_{1},a_{2},...,a_{n})=1}$ 

(ii) ${\displaystyle {a_{1}+a_{2}+...+a_{n}=0}}$ 

(iii) no proper subsum of ${\displaystyle {a_{1},a_{2},...,a_{n}}}$  equals ${\displaystyle {0}}$ 

First formulation

The n conjecture states that for every ${\displaystyle {\varepsilon >0}}$ , there is a constant ${\displaystyle C}$ , depending on ${\displaystyle {n}}$  and ${\displaystyle {\varepsilon }}$ , such that:

${\displaystyle \operatorname {max} (|a_{1}|,|a_{2}|,...,|a_{n}|)<C_{n,\varepsilon }\operatorname {rad} (|a_{1}|\cdot |a_{2}|\cdot ...\cdot |a_{n}|)^{2n-5+\varepsilon }}$ 

where ${\displaystyle \operatorname {rad} (m)}$  denotes the radical of the integer ${\displaystyle {m}}$ , defined as the product of the distinct prime factors of ${\displaystyle {m}}$ .

Second formulation

Define the quality of ${\displaystyle {a_{1},a_{2},...,a_{n}}}$  as

${\displaystyle q(a_{1},a_{2},...,a_{n})={\frac {\log(\operatorname {max} (|a_{1}|,|a_{2}|,...,|a_{n}|))}{\log(\operatorname {rad} (|a_{1}|\cdot |a_{2}|\cdot ...\cdot |a_{n}|))}}}$ 

The n conjecture states that ${\displaystyle \limsup q(a_{1},a_{2},...,a_{n})=2n-5}$ .

Stronger form

Vojta (1998) proposed a stronger variant of the n conjecture, where setwise coprimeness of ${\displaystyle {a_{1},a_{2},...,a_{n}}}$  is replaced by pairwise coprimeness of ${\displaystyle {a_{1},a_{2},...,a_{n}}}$ .

There are two different formulations of this strong n conjecture.

Given ${\displaystyle {n\geq 3}}$ , let ${\displaystyle {a_{1},a_{2},...,a_{n}\in \mathbb {Z} }}$  satisfy three conditions:

(i) ${\displaystyle {a_{1},a_{2},...,a_{n}}}$  are pairwise coprime

(ii) ${\displaystyle {a_{1}+a_{2}+...+a_{n}=0}}$ 

(iii) no proper subsum of ${\displaystyle {a_{1},a_{2},...,a_{n}}}$  equals ${\displaystyle {0}}$ 

First formulation

The strong n conjecture states that for every ${\displaystyle {\varepsilon >0}}$ , there is a constant ${\displaystyle C}$ , depending on ${\displaystyle {n}}$  and ${\displaystyle {\varepsilon }}$ , such that:

${\displaystyle \operatorname {max} (|a_{1}|,|a_{2}|,...,|a_{n}|)<C_{n,\varepsilon }\operatorname {rad} (|a_{1}|\cdot |a_{2}|\cdot ...\cdot |a_{n}|)^{1+\varepsilon }}$ 

Second formulation

Define the quality of ${\displaystyle {a_{1},a_{2},...,a_{n}}}$  as

${\displaystyle q(a_{1},a_{2},...,a_{n})={\frac {\log(\operatorname {max} (|a_{1}|,|a_{2}|,...,|a_{n}|))}{\log(\operatorname {rad} (|a_{1}|\cdot |a_{2}|\cdot ...\cdot |a_{n}|))}}}$ 

The strong n conjecture states that ${\displaystyle \limsup q(a_{1},a_{2},...,a_{n})=1}$ .

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". arXiv:math/9806171. Bibcode:1998math......6171V. {{cite journal}}: Cite journal requires |journal= (help)