SE-SKOLORNAS 1975 Problem 2

Is there a positive integer $n$ such that the fractional part of $\left(3+\sqrt{5}\right)^n >0.99 ?$

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.