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n conjecture

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Generalization of the abc conjecture to more than three integers
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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 n ≥ 3 {\displaystyle n\geq 3} , let a 1 , a 2 , . . . , a n ∈ Z {\displaystyle a_{1},a_{2},...,a_{n}\in \mathbb {Z} } satisfy three conditions:

(i) gcd ( a 1 , a 2 , . . . , a n ) = 1 {\displaystyle \gcd(a_{1},a_{2},...,a_{n})=1}
(ii) a 1 + a 2 + . . . + a n = 0 {\displaystyle a_{1}+a_{2}+...+a_{n}=0}
(iii) no proper subsum of a 1 , a 2 , . . . , a n {\displaystyle a_{1},a_{2},...,a_{n}} equals 0 {\displaystyle 0}

First formulation

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

max ⁡ ( | a 1 | , | a 2 | , . . . , | a n | ) < C n , ε rad ⁡ ( | a 1 | ⋅ | a 2 | ⋅ … ⋅ | a n | ) 2 n − 5 + ε {\displaystyle \operatorname {max} (|a_{1}|,|a_{2}|,...,|a_{n}|)<C_{n,\varepsilon }\operatorname {rad} (|a_{1}|\cdot |a_{2}|\cdot \ldots \cdot |a_{n}|)^{2n-5+\varepsilon }}

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

Second formulation

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

q ( a 1 , a 2 , . . . , a n ) = log ⁡ ( max ⁡ ( | a 1 | , | a 2 | , . . . , | a n | ) ) log ⁡ ( rad ⁡ ( | a 1 | ⋅ | a 2 | ⋅ . . . ⋅ | a n | ) ) {\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 lim sup q ( a 1 , a 2 , . . . , a n ) = 2 n − 5 {\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 a 1 , a 2 , . . . , a n {\displaystyle a_{1},a_{2},...,a_{n}} is replaced by pairwise coprimeness of a 1 , a 2 , . . . , a n {\displaystyle a_{1},a_{2},...,a_{n}} .

There are two different formulations of this strong n conjecture.

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

(i) a 1 , a 2 , . . . , a n {\displaystyle a_{1},a_{2},...,a_{n}} are pairwise coprime
(ii) a 1 + a 2 + . . . + a n = 0 {\displaystyle a_{1}+a_{2}+...+a_{n}=0}
(iii) no proper subsum of a 1 , a 2 , . . . , a n {\displaystyle a_{1},a_{2},...,a_{n}} equals 0 {\displaystyle 0}

First formulation

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

max ⁡ ( | a 1 | , | a 2 | , . . . , | a n | ) < C n , ε rad ⁡ ( | a 1 | ⋅ | a 2 | ⋅ … ⋅ | a n | ) 1 + ε {\displaystyle \operatorname {max} (|a_{1}|,|a_{2}|,...,|a_{n}|)<C_{n,\varepsilon }\operatorname {rad} (|a_{1}|\cdot |a_{2}|\cdot \ldots \cdot |a_{n}|)^{1+\varepsilon }}

Second formulation

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

q ( a 1 , a 2 , . . . , a n ) = log ⁡ ( max ⁡ ( | a 1 | , | a 2 | , . . . , | a n | ) ) log ⁡ ( rad ⁡ ( | a 1 | ⋅ | a 2 | ⋅ . . . ⋅ | a n | ) ) {\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 lim sup q ( a 1 , a 2 , . . . , a n ) = 1 {\displaystyle \limsup q(a_{1},a_{2},...,a_{n})=1} .

References

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