Diophantine triples with largest two elements in common
In this paper we prove that if { a , b , c } is a Diophantine triple with a < b < c , then { a + 1 , b , c } cannot be a Diophantine triple. Moreover, we show that if { a 1 , b , c } and { a 2 , b , c } are Diophantine triples with a 1 < a 2 < b < c < 16 b 3 , then { a 1 , a 2 , b...
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| Veröffentlicht in: | Periodica mathematica Hungarica 2021-03, Vol.82 (1), p.56-68 |
|---|---|
| Hauptverfasser: | , , |
| Format: | Artikel |
| Sprache: | eng |
| Schlagworte: | |
| Online-Zugang: | Volltext |
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| Zusammenfassung: | In this paper we prove that if
{
a
,
b
,
c
}
is a Diophantine triple with
a
<
b
<
c
, then
{
a
+
1
,
b
,
c
}
cannot be a Diophantine triple. Moreover, we show that if
{
a
1
,
b
,
c
}
and
{
a
2
,
b
,
c
}
are Diophantine triples with
a
1
<
a
2
<
b
<
c
<
16
b
3
, then
{
a
1
,
a
2
,
b
,
c
}
is a Diophantine quadruple. In view of these results, we conjecture that if
{
a
1
,
b
,
c
}
and
{
a
2
,
b
,
c
}
are Diophantine triples with
a
1
<
a
2
<
b
<
c
, then
{
a
1
,
a
2
,
b
,
c
}
is a Diophantine quadruple. |
|---|---|
| ISSN: | 0031-5303 1588-2829 |
| DOI: | 10.1007/s10998-020-00331-4 |