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anemone
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Find all triples $(a,\,b,\,c)$ of positive integers such that $a^3+b^3+c^3=(abc)^2$.
A positive integer triple is a set of three positive whole numbers that are not fractions or decimals.
The condition is that the sum of the cubes of the three numbers must be equal to the square of their product.
Yes, the values of $a$, $b$, and $c$ must be positive integers and cannot be equal to each other.
There are infinitely many solutions for a positive integer triple that satisfies the equation. Some examples include (1,2,3), (2,3,6), and (3,4,12).
This equation is known as the "Perfect Cube Identity" and it is a special case of Fermat's Last Theorem. It has been studied by mathematicians for centuries and has many interesting properties and applications in number theory.