Find 3 positive integers and which satisfy this equation:

Using this solution, find a different solution by changing one and only one of the numbers.

How many solutions does the equation have?

Source:

http://galileo.org/classroom-examples/math/math-fair-problems/puzzles/xyz-tree/

Extensions:

Are all triple solutions which contain a number, , connected to each other on the tree found below?

Adding some triples to the tree:

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,

,

,

,

,

,

,

,

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the next values for are