Does the Diophantine equation

have an infinite number of relatively prime solutions?

Solution:

we have the trivial solutions and permutations thereof.

For other solutions, note that each of the three equations

is satisfied by

consequently, we have the infinite set of solutions

where

I don’t know whether or not there are any other infinite sets of relatively prime solutions.

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Yes, if you look at these few solutions you can see the pattern.

{1,4,9}

{1,9,16}

{1,16,25}

{1,25,36}

{1,36,49}

{1,49,64}

{1,64,81}

{4,9,25}

{4,25,49}

{4,49,81}

{9,16,49}

{9,25,64}

{16,25,81}

I posted a general solution.