A generalization of a theorem of Baker and Davenport

Abstract

The Greek mathematician Diophantus of Alexandria 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 [4]). The first set of four positive integers with the above property was found by Fermat, and it was {1, 3, 8, 120}. A set of positive integers {a1, a2, . . . , am} is said to have the property of Diophantus if aiaj +1 is a perfect square for all 1 ≤ i < j ≤ m. Such a set is called a Diophantine m-tuple (or P1-set of size m). In 1969, Baker and Davenport [2] proved that if d is a positive integer such that {1, 3, 8, d} is a Diophantine quadruple, then d has to be 120. The same result was proved by Kanagasabapathy and Ponnudurai [9], Sansone [12] and Grinstead [7]. This result implies that the Diophantine triple {1, 3, 8} cannot be extended to a Diophantine quintuple. In the present paper we generalize the result of Baker and Davenport and prove that the Diophantine pair {1, 3} can be extended to infinitely many Diophantine quadruples, but it cannot be extended to a Diophantine quintuple.