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Diophantine quintuple conjecture
Definition A set of m positive integers
is called a Diophantine
-tuple if
is a perfect square for all
.
![$ \{a_1, a_2, \dots, a_m\} $](/files/tex/301be6969a88d80d573c770feb4f3eb1f8e6a561.png)
![$ m $](/files/tex/ddaab6dc091926fb1da549195000491cefae85c1.png)
![$ a_i\cdot a_j + 1 $](/files/tex/01b039b5827cce47134f50a91e5ed5cc95f610da.png)
![$ 1 \leq i < j \leq m $](/files/tex/7ee028ce5233e7eaa33d3c4e090e0be23eadfdf5.png)
Conjecture (1) Diophantine quintuple does not exist.
It would follow from the following stronger conjecture [Da]:
Conjecture (2) If
is a Diophantine quadruple and
, then
![$ \{a, b, c, d\} $](/files/tex/6f295c1b141b37f265f8228f947cabbbd744231b.png)
![$ d > \max \{a, b, c\} $](/files/tex/02dc84e05b3bf5eff57f8e539e336d62abb44f5c.png)
![$ d = a + b + c + 2bc + 2\sqrt{(ab+1)(ac+1)(bc+1)}. $](/files/tex/ad227c1e1f4b4ca92da7fe0a897a799ad9c1b88d.png)
It was proved in [Db] that there are only finitely many Diophantine quintuples and no Diophantine sextuples.
Conjecture (2) is motivated by an observation of [AHS] that every Diophantine triple can be extended to a Diophantine quadruple
Bibliography
[Da] A. Dujella Diophantine -tuples, a survey of the main problems and results concerning Diophantine m-tuples.
[Db] A. Dujella, There are only finitely many Diophantine quintuples, J. Reine Angew. Math. 566 (2004), 183-214.
[AHS] J. Arkin, V. E. Hoggatt and E. G. Strauss, On Euler's solution of a problem of Diophantus, Fibonacci Quart. 17 (1979), 333-339.
* indicates original appearance(s) of problem.
This result has been proven
in a paper announced in 2016 and published in 2019, He, Togbé and Ziegler [350] gave the proof of the Diophantine quintuple conjecture