A Problem of Diophantus and Pell Numbers

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 its any two distinct elements increased by 1 is a square of a rational number (see [3]). Fermat first found a set of four positive integers with the above property, and it was {1, 3, 8, 120}. In 1969, Davenport and Baker [2] showed that if positive integers 1, 3, 8 and d have this property then d must be 120. Let n be an integer. A set of positive integers {a1, a2, . . . , am} is said to have the property of Diophantus of order n, symbolically D(n), if aiaj + n is a perfect square for all 1 ≤ i < j ≤ m. The sets with the property D(l2) were particularly discussed in [4]. It was proved that for any integer l and any set {a, b} with the property D(l2), where ab is not a perfect square, there exist an infinite number of sets of the form {a, b, c, d} with the property D(l2). This result is the generalization of well known result for l = 1 (see [8]). The proof of this result is based on the construction of the double sequences yn,m and zn,m which are defined in [4] by second order recurrences in each indices. Solving these recurrences we obtain