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
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}
.
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}
.
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 . abc conjecture&rft.volume=1998&rft.issue=21&rft.pages=1103-1116&rft.date=1998&rft_id=info:arxiv/math/9806171&rft_id=https://mathscinet.ams.org/mathscinet-getitem?mr=1663215#id-name=MR&rft_id=info:doi/10.1155/S1073792898000658&rft.aulast=Vojta&rft.aufirst=Paul&rfr_id=info:sid/en.wikipedia.org:N conjecture" class="Z3988">{{cite journal }}
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