A Note on Diophantine Quintuples

Introduction. Diophantus noted that the rational numbers 1/16, 33/16, 17/4 and 105/16 have the following property: the product of any two of them increased by 1 is a square of a rational number (see [2, 3]). Let n be an integer. A set of positive integers {a1, a2, . . . , am} is said to have the property D(n) if aiaj + n is a perfect square for all 1 ≤ i < j ≤ m. Such a set is called a Diophantine m-tuple. Fermat first found an example of a Diophantine quadruple with the property D(1), and it was {1, 3, 8, 120} (see [2]). In 1985, Brown [1], Gupta and Singh [7] and Mohanty and Ramasamy [9] proved independently that if n ≡ 2 (mod 4), then there does not exist a Diophantine quadruple with the property D(n). If n 6≡ 2 (mod 4) and n 6∈ {−4,−3,−1, 3, 5, 8, 12, 20}, then there exists at least one Diophantine quadruple with the property D(n) (see [4, Theorem 5]. In [5], the definition of Diophantine m-tuples is extended to the rational numbers. Namely, if q is a rational number, the set of non-zero rationals {a1, a2, . . . , am} is called a rational Diophantine m-tuple with the property D(q) if aiaj + q is a square of a rational number for all 1 ≤ i < j ≤ m.