AT-OEMO-FED1 2011 Problem 1

Determine all integer triplets $(x,y,z)$ such that $x^4+x^2=7^zy^2\text{.}$

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.