论文标题
在$ d（4）$ - Pairs $ \ {a，ka \} $带有$ k \ in \ in \ {2,3,6 \} $中
On the $D(4)$-pairs $\{a, ka\}$ with $k\in \{2,3,6\}$
论文作者
论文摘要
令$ a $和$ b = ka $为正整数，$ k \ in \ {2、3、6 \}，$，使得$ ab+4 $是一个完美的正方形。在本文中，我们研究了$ d（4）$ - 对$ \ {a，ka \}的可扩展性。$更确切地说，我们证明，考虑到三个积极整数$ c $的家庭，取决于$ a，$ a，$ \ {a，b，b，b，b，c，d \} $是$ a的$ n e Enter the Offer the Offer the en e Enter the Onder the Onder the Onder Ond e Ens the Orter Ond e e Ens the Orter Ond e e Ens of Botern us n e e e e eys Onder的产品。 $$ d = a+b+c+\ frac {1} {2} \ left（abc \ pm \ pm \ sqrt {（ab+4）（ac+4）（ac+4）（bc+4）} \ right）。$ d $。
Let $a$ and $b=ka$ be positive integers with $k\in \{2, 3, 6\},$ such that $ab+4$ is a perfect square. In this paper, we study the extensibility of the $D(4)$-pairs $\{a, ka\}.$ More precisely, we prove that by considering three families of positive integers $c$ depending on $a,$ if $\{a, b, c, d\}$ is the set of positive integers which has the property that the product of any two of its elements increased by $4$ is a perfect square, then $d$ in given by $$d=a+b+c+\frac{1}{2}\left(abc\pm \sqrt{(ab+4)(ac+4)(bc+4)}\right).$$ As a corollary, we prove that any $D(4)$-quadruple which contains the pair $\{a, ka\}$ is regular.
