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How many different natural numbers are there such that the difference of any two is a perfect square?

I could find that with 'sums' instead of 'differences' this has been asked by Erdos and L. Moser (see Guy: Unsolved Problems in Number Theory, 3rd ed., p. 268), and that is open, with six being the current record.

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    $\begingroup$ Just one unless you consider only the positive differences. $\endgroup$
    – W-t-P
    Commented Oct 19, 2020 at 20:23

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Four is possible, an example is $0,451584,462400,485809$. This solution comes from the example of a cuboid with integer sides, space diagonal and two out of three face diagonals given here. I don't know for $5$ or more but it is probably a very hard question.

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