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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?

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  • $\begingroup$ 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. $\endgroup$
    – MJD
    Mar 19, 2015 at 1:41
  • $\begingroup$ Oh, thanks, I might. But I mostly got the response I wanted, thanks to you! $\endgroup$
    – Brian Tung
    Mar 19, 2015 at 17:38

2 Answers 2

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

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  • $\begingroup$ 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. $\endgroup$
    – Brian Tung
    Mar 18, 2015 at 21:52
  • $\begingroup$ 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. $\endgroup$
    – MJD
    Mar 18, 2015 at 21:58
  • $\begingroup$ 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! $\endgroup$
    – Brian Tung
    Mar 18, 2015 at 22:04
  • $\begingroup$ @MJD: Would you mind sharing your code? $\endgroup$
    – Tebbe
    Mar 19, 2015 at 18:01
  • $\begingroup$ @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. $\endgroup$
    – MJD
    Mar 19, 2015 at 18:05
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$$ a=3, b=4, c=3, d=4........... $$

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  • $\begingroup$ I'm wondering if there exist pairwise coprime integers satisfying the question. $\endgroup$
    – William Stagner
    Mar 18, 2015 at 21:43
  • $\begingroup$ I certainly didn't downvote it (and would not). I've edited the question to eliminate that kind of answer. $\endgroup$
    – Brian Tung
    Mar 18, 2015 at 21:54

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