ACTA ARITHMETICA 136.2 (2009) An irregular D(4)-quadruple cannot be extended to a quintuple by Alan Filipin (Zagreb) 1. Introduction Definition 1. Let n be an integer. A set of m positive integers is called a Diophantine m-tuple with the property D(n), or simply D(n)-m-tuple , if the product of any two of them increased by n is a perfect square. Diophantus was the first to look for such sets in the case n =1. He found a set of four positive rational numbers with the above property:  1 16 , 33 16 , 17 4 , 105 16  . Fermat found a first D(1)-quadruple, the set {1, 3, 8, 120}. Euler was later able to add the fifth positive rational, 777480 8288641 , to Fermat’s set (see [3], [4, pp. 103–104, 232]). Recently, Gibbs [15] found several ex- amples of D(n)-sextuples, e.g. {99, 315, 9920, 32768, 44460, 19534284} is a D(2985984)-sextuple. There is a folklore conjecture that there does not exist a D(1)-quintuple. The first result supporting this conjecture is due to Baker and Davenport [1], who proved that Fermat’s set cannot be extended to a D(1)-quintuple. Dujella [7] proved that there does not exist a D(1)-sextuple and that there are only finitely many D(1)-quintuples. Considering congru- ences modulo 8, it is easy to prove that a D(4)-m-tuple can contain at most two odd numbers. So Dujella’s result implies that there does not exist a D(4)-8-tuple and that there are only finitely many D(4)-septuples (see [9]). The author [11, 12] improved this result by proving that there does not exist a D(4)-sextuple. In the present paper we further improve this result. For n = 4 it is conjectured that there does not exist a D(4)-quintuple. Actually, there is even a stronger version of that conjecture. Conjecture 1 (cf. [9, Conjecture 1]). There does not exist a D(4)- quintuple. Moreover , if {a, b, c, d} is a D(4)-quadruple such that a<b< c < d, then d = a + b + c + 1 2 (abc + rst), 2000 Mathematics Subject Classification : 11D09, 11D45. Key words and phrases : Diophantine m-tuples. DOI: 10.4064/aa136-2-5 [167] c  Instytut Matematyczny PAN, 2009