To make this more precise, we are looking for four (ETA: distinct) positive integers $a$, $b$, $c$, and $d$, such that $\sqrt{a^2+b^2}$, $\sqrt{b^2+c^2}$, $\sqrt{c^2+d^2}$, and $\sqrt{d^2+a^2}$ are all integers as well.

Equivalently, we seek a convex quadrilateral with integer sides, whose diagonals intersect at right angles at a point a (ETA: distinct) integer distance from all four vertices.

ETA: Answered in the affirmative below, by computer search. Is there a more elegant, less brute-forcey way to such an answer?

  • I think you will get better results if you post a fresh question. Don't forget this time to say that the numbers should be distinct. – MJD Mar 19 '15 at 1:41
  • Oh, thanks, I might. But I mostly got the response I wanted, thanks to you! – Brian Tung Mar 19 '15 at 17:38
up vote 5 down vote accepted

Computer search finds many examples; considering only those where all four numbers are distinct, we have for example:

$$\begin{align} a & = 6375 \\ b& = 6512 \\ c & = 9984 \\d & = 800 \end{align}$$

and $$\begin{align} a & = 3472 \\ b& = 7296 \\ c & = 10400 \\d & = 2175 \end{align}$$

  • Brilliant, thanks. (Sincerely.) I did run a search, but didn't extend it far enough. Also, I was curious whether there was better than a brute-force approach to the question. – Brian Tung Mar 18 '15 at 21:52
  • My approach was a brute force search over all pythagorean triples. this is much faster than a search over all integers; It took a fraction of a second to run on my laptop. If you're interested I'll be glad to explain in more detail. – MJD Mar 18 '15 at 21:58
  • No, I think I get the idea. I was lazy previously. But as you say, such a search, though not going through all quadruples, is still brute force. Nothing wrong particularly about that, but I was hoping for something with a bit more analytical flavor to it. Thanks for offering the answer so quickly! – Brian Tung Mar 18 '15 at 22:04
  • @MJD: Would you mind sharing your code? – Tebbe Mar 19 '15 at 18:01
  • @Tebbe github.com/mjdominus/perl-misc/blob/master/math/pt is the searcher, and github.com/mjdominus/perl-misc/blob/master/math/pttest is a checker that validates that the solutions printed by the first program are actually correct. – MJD Mar 19 '15 at 18:05

$$ a=3, b=4, c=3, d=4........... $$

  • I'm wondering if there exist pairwise coprime integers satisfying the question. – William Stagner Mar 18 '15 at 21:43
  • I certainly didn't downvote it (and would not). I've edited the question to eliminate that kind of answer. – Brian Tung Mar 18 '15 at 21:54

Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Not the answer you're looking for? Browse other questions tagged or ask your own question.