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
, let
a
1
,
a
2
,
.
.
.
,
a
n
∈
Z
satisfy three conditions:
(i)
gcd
(
a
1
,
a
2
,
.
.
.
,
a
n
)
=
1
(ii)
a
1
+
a
2
+
.
.
.
+
a
n
=
0
(iii) no proper subsum of
a
1
,
a
2
,
.
.
.
,
a
n
equals
0
First formulation
The n conjecture states that for every
ε
>
0
, there is a constant
C
, depending on
n
and
ε
, such that:
max
(
|
a
1
|
,
|
a
2
|
,
.
.
.
,
|
a
n
|
)
<
C
n
,
ε
rad
(
|
a
1
|
⋅
|
a
2
|
⋅
.
.
.
⋅
|
a
n
|
)
2
n
−
5
+
ε
where
rad
(
m
)
denotes the radical of the integer
m
, defined as the product of the distinct prime factors of
m
.
Second formulation
Define the quality of
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
|
)
)
The n conjecture states that
lim sup
q
(
a
1
,
a
2
,
.
.
.
,
a
n
)
=
2
n
−
5
.
Vojta (1998) proposed a stronger variant of the n conjecture, where setwise coprimeness of
a
1
,
a
2
,
.
.
.
,
a
n
is replaced by pairwise coprimeness of
a
1
,
a
2
,
.
.
.
,
a
n
.
There are two different formulations of this strong n conjecture.
Given
n
≥
3
, let
a
1
,
a
2
,
.
.
.
,
a
n
∈
Z
satisfy three conditions:
(i)
a
1
,
a
2
,
.
.
.
,
a
n
are pairwise coprime
(ii)
a
1
+
a
2
+
.
.
.
+
a
n
=
0
(iii) no proper subsum of
a
1
,
a
2
,
.
.
.
,
a
n
equals
0
First formulation
The strong n conjecture states that for every
ε
>
0
, there is a constant
C
, depending on
n
and
ε
, such that:
max
(
|
a
1
|
,
|
a
2
|
,
.
.
.
,
|
a
n
|
)
<
C
n
,
ε
rad
(
|
a
1
|
⋅
|
a
2
|
⋅
.
.
.
⋅
|
a
n
|
)
1
+
ε
Second formulation
Define the quality of
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
|
)
)
The strong n conjecture states that
lim sup
q
(
a
1
,
a
2
,
.
.
.
,
a
n
)
=
1
.
