Loading…
On integer values of sum and product of three positive rational numbers
In 1997 we proved that if n is of the form 4 k , 8 k - 1 or 2 2 m + 1 ( 2 k - 1 ) + 3 , where k , m ∈ N , then there are no positive rational numbers x , y , z satisfying x y z = 1 , x + y + z = n . Recently, N. X. Tho proved the following statement: let a ∈ N be odd and let either n ≡ 0 ( mod 4 )...
Saved in:
| Published in: | Periodica mathematica Hungarica 2023-12, Vol.87 (2), p.484-497 |
|---|---|
| Main Author: | |
| Format: | Article |
| Language: | English |
| Subjects: | |
| Citations: | Items that this one cites |
| Online Access: | Get full text |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| Summary: | In 1997 we proved that if
n
is of the form
4
k
,
8
k
-
1
or
2
2
m
+
1
(
2
k
-
1
)
+
3
,
where
k
,
m
∈
N
, then there are no positive rational numbers
x
,
y
,
z
satisfying
x
y
z
=
1
,
x
+
y
+
z
=
n
.
Recently, N. X. Tho proved the following statement: let
a
∈
N
be odd and let either
n
≡
0
(
mod
4
)
or
n
≡
7
(
mod
8
)
. Then the system of equations
x
y
z
=
a
,
x
+
y
+
z
=
a
n
.
has no solutions in positive rational numbers
x
,
y
,
z
. A representative example of our result is the following statement: assume that
a
,
n
∈
N
are such that at least one of the following conditions holds:
n
≡
0
(
mod
4
)
n
≡
7
(
mod
8
)
a
≡
0
(
mod
4
)
a
≡
0
(
mod
2
)
and
n
≡
3
(
mod
4
)
a
2
n
3
=
2
2
m
+
1
(
2
k
-
1
)
+
27
for some
k
,
m
∈
N
.
Then the system of equations
x
y
z
=
a
,
x
+
y
+
z
=
a
n
.
has no solutions in positive rational numbers
x
,
y
,
z
. |
|---|---|
| ISSN: | 0031-5303 1588-2829 |
| DOI: | 10.1007/s10998-023-00529-2 |