n conjecture

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

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 ${\displaystyle \limsup q(a_{1},a_{2},...,a_{n})=1}$.