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.


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.

  • $\begingroup$ Why do you believe this to be the case? Is there some context for the problem? Have you tried any examples? $\endgroup$
    – lulu
    Aug 14, 2022 at 12:08
  • $\begingroup$ @lulu edited...... $\endgroup$
    – abcdefu
    Aug 14, 2022 at 12:14
  • $\begingroup$ Can you provide some references? I have not seen this problem before... $\endgroup$
    – lulu
    Aug 14, 2022 at 12:16
  • $\begingroup$ For $(3,5,16)$ one solution is $d=1008$. Interesting that one needs to go out so far... $\endgroup$
    – lulu
    Aug 14, 2022 at 12:19
  • $\begingroup$ @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 $\endgroup$
    – abcdefu
    Aug 14, 2022 at 12:20

1 Answer 1


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.)

  • $\begingroup$ so this means that there is no exact proof given by Arkin, Hoggatt and Strauss$?$ $\endgroup$
    – abcdefu
    Aug 14, 2022 at 16:55
  • $\begingroup$ Yes they proved it; edited to clarify $\endgroup$ Aug 14, 2022 at 16:56
  • $\begingroup$ do you know the proof they gave? $\endgroup$
    – abcdefu
    Aug 14, 2022 at 16:58
  • 1
    $\begingroup$ see arxiv.org/html/math/9903035. This terminology should give you the tools you need to learn more on your own. $\endgroup$ Aug 14, 2022 at 17:02
  • 1
    $\begingroup$ @Greg Martin.-If you hadn't given your answer, I would have spent a lot of time (in vain, I think) trying to find one. Thanks. $\endgroup$
    – Piquito
    Aug 14, 2022 at 17:12

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