A generalization of a theorem of Baker and Davenport

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.