N conjecture

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

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 ...\cdot |a_{n}|)^{2n-5+\varepsilon }}

where rad ⁡ ( m ) {\displaystyle \operatorname {rad} (m)} denotes the radical of the 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 ...\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

  • 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)