# Is there a positive integer $d$ in existence$?$

Suppose there are three positive integers $$a,b,c$$ such that the product of any two is one less than an integer squared, i.e., \begin{align} ab+1=x^2\\ ac+1=y^2\\ bc+1=z^2 \end{align} Then is it true that there exists a fourth positive integer $$d$$ such that its product with any of the first three is also one less than a square, i.e. \begin{align} ad+1=w^2\\ bd+1=u^2\\ cd+1=v^2 \end{align}

I tried factorizing the expressions but it did not prove to be of much help. Like we can say that $$ab=(x+1)(x-1)$$ and similarly for all expressions. Any help is greatly appreciated.

EDIT

A triplet satisfying $$(a,b,c)$$ is $$(1,3,8)$$ and the value of $$d$$ in this case is $$120$$. As far as I know, this is an old and very interesting problem and many famous mathematicians had attacked this problem but I guess none of them gave a detailed proof or maybe I don't know about it.

• Why do you believe this to be the case? Is there some context for the problem? Have you tried any examples?
– lulu
Aug 14, 2022 at 12:08
• @lulu edited...... Aug 14, 2022 at 12:14
• Can you provide some references? I have not seen this problem before...
– lulu
Aug 14, 2022 at 12:16
• For $(3,5,16)$ one solution is $d=1008$. Interesting that one needs to go out so far...
– lulu
Aug 14, 2022 at 12:19
• @lulu well, i have seen this problem while I was surfing through internet but unfortunately i forgot to save the link....that's why I asked it here Aug 14, 2022 at 12:20

It turns out that this is a classical topic in number theory. In the terminology you're seeking, you're asking whether every "Diophantine triple" of positive integers can be extended to a "Diophantine quadruple" of positive integers. And the answer is yes: Arkin, Hoggatt, and Strauss proved in 1979 that (in your notation) you can choose $$d = a + b + c + 2abc + 2xyz.$$ (It was a well-known conjecture in this field, for example, that there do not exist any Diophantine quintuples of positive integers; this was only proved recently, by He, Togbé, and Ziegler in 2019.)
• so this means that there is no exact proof given by Arkin, Hoggatt and Strauss$?$ Aug 14, 2022 at 16:55