Joel's Problem 5

Number Theory Level pending

Suppose that in a set \(A\) of mutually coprime positive integers, for any ordered triple \((x, y, z)\) in \(A\) with \(x, y, z\) mutually distinct, \(x^{2}\) is a factor of \(y^{3}+z^{3}\). Let \(X\) be the number of distinct elements in \(A\). Find the maximum possible value of \(X\).

(If \(X<3\), \((x, y, z)\) might not exist.)

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