On Diophantine quintuples

The Greek mathematician Diophantus of Alexandria noted that the set { 1 16 , 33 16 , 17 4 , 105 16 } has the following property: the product of any two of its distinct elements increased by 1 is a square of a rational number (see [5]). Fermat first found a set of four positive integers with the above property, and it was {1, 3, 8, 120}. Let n be an integer. A set of positive integers {x1, x2, . . . , xm} is said to have the property D(n) if for all 1 ≤ i < j ≤ m the following holds: xixj + n = y 2 ij , where yij is an integer. Such a set is called a Diophantine m-tuple. Davenport and Baker [4] showed that if d is a positive integer such that the set {1, 3, 8, d} has the property of Diophantus, then d has to be 120. This implies that the Diophantine quadruple {1, 3, 8, 120} cannot be extended to the Diophantine quintuple with the propertyD(1). Analogous result was proved for the Diophantine quadruple {2, 4, 12, 420} with the property D(1) [17], for the Diophantine quadruple {1, 5, 12, 96} with the propertyD(4) [15] and for the Diophantine quadruples {k−1, k+1, 4k, 16k3−4k} with the property D(1) for almost all positive integers k [9]. Euler proved that every Diophantine pair {x1, x2} with the property D(1) can be extended in infinitely many ways to the Diophantine quadruple with the same property (see [12]). In [6] it was proved that the same conclusion is valid for the pair with the property D(l2) if the additional condition that x1x2 is not a perfect square is fulfilled. Arkin, Hoggatt and Strauss [3] proved that every Diophantine triple with the property D(1) can be extended to the Diophantine quadruple. More precisely, if xixj + 1 = y 2 ij , then we can set x4 = x1 + x2 + x3 + 2x1x2x3+2y12y13y23. For the Diophantine quadruple obtained in this way, they proved the existence of a positive rational number x5 with the property that xix5 + 1 is a square of a rational number for i = 1, 2, 3, 4. Using this construction, in [2, 7, 8, 11] some formulas for Diophantine