On the \(D(4)\)-pairs \(\{a, ka\}\) with \(k\in \{2,3,6\}\)

Let \(a\) and \(b=ka\) be positive integers with \(k\in \{2, 3, 6\},\) 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±√((ab+4)(ac+4)(bc+4))). As a corollary, we prove that any \(D(4)\)-quadruple tht contains the pair \(\{a, ka\}\) is regular.