KZ-MO 2000 Problem 4

Find all triples of natural numbers $(x, y, z)$ that satisfy the condition $(x + 1) ^ {y + 1} + 1 = (x + 2) ^ {z + 1}.$

Dataset text AI-MO AoPS. Quoted with source.

Solutions

No solution is on olymmath yet. Official and human solutions come first when they arrive. AI solutions come after them, with a label.