Let n be a nonzero integer. A set of m positive integers {a1, a2, …, am} is said to
have the property D(n) if aiaj+n is a perfect square for all 1 [les ] i [les ] j [les ] m. Such a
set is called a Diophantine m-tuple (with the property D(n)), or Pn-set of size m.Diophantus found the quadruple {1, 33, 68, 105} with the property D(256). The
first Diophantine quadruple with the property D(1), the set {1, 3, 8, 120}, was found
by Fermat (see [8, 9]). Baker and Davenport [3] proved that this Fermat’s set cannot
be extended to the Diophantine quintuple, and a famous conjecture is that there does
not exist a Diophantine quintuple with the property D(1). The theorem of Baker and
Davenport has been recently generalized to several parametric families of quadruples
[12, 14, 16], but the conjecture is still unproved.On the other hand, there are examples of Diophantine quintuples and sextuples like
{1, 33, 105, 320, 18240} with the property D(256) [11] and {99, 315, 9920,
32768, 44460, 19534284} with the property D(2985984) [19]].
Diophantine quintuples containing triples of the first kind
