Integer solutions to x + y^2 + z^3 = 50

Since $x, y, z \ge 1$ are positive integers:

• Case $z = 1$: $z^3 = 1 \implies x + y^2 = 49$. Since $y^2 < 49$, $y \in \{1, 2, 3, 4, 5, 6\}$, giving $6$ solutions for $x$.
• Case $z = 2$: $z^3 = 8 \implies x + y^2 = 42$. $y^2 < 42 \implies y \in \{1, 2, 3, 4, 5, 6\}$, giving $6$ solutions for $x$.
• Case $z = 3$: $z^3 = 27 \implies x + y^2 = 23$. $y^2 < 23 \implies y \in \{1, 2, 3, 4\}$, giving $4$ solutions for $x$.
• Case $z \ge 4$: $z^3 \ge 64 > 50$, no positive integer solutions.

Total solutions $= 6 + 6 + 4 = 16$.