ON THE D(4)-PAIRS {a, ka} WITH k ? {2,3,6}

Let a and b = ka be positive integers with k. {2, 3, 6}, such that ab+4 is a perfect square. In this paper, we study the extensibility of the D(4)-pairs {a, ka}. More precisely, we prove that by considering families of positive integers c depending on a, if {a, b, c, d} is a set of positive integers which has the property that the product of any two of its elements increased by 4 is a perfect square, then d is given by d = a + b + c + 1 2 abc +/- p (ab + 4)(ac + 4)(bc + 4). As a corollary, we prove that any D(4)-quadruple tht contains the pair {a, ka} is regular.