Find highest power of 2 that divides $3^{2^k}-1$

Question

I am trying to find highest power of 2 that divides $3^{2^k}-1$ but I have no idea where to start - could you give me any hint?

Answer 1

Hints: $3^{2}-1 = 8,$ and $3^{2^{h}}-1 = (3^{2^{h-1}}-1)(3^{2^{h-1}}+1)$ for $h >1.$ Compare the power of $2$ dividing each factor on the right hand side of the last expression.