Proof that $3^n | 2^{3^n} + 1$ Question:

Proof by induction that $3^n | 2^{3^n} + 1$. 

Attempt: $$ 2^{3^{n+1}} + 1 = 2^{3^n}  2^3 + 1 = 2^{3^n}  2^3 + 1 + 2^3 - 2^3 = 
 2^3( 2^{3^n} + 1 ) + 1 -2^3$$
And the first is $3^n |$ but second I don't know how to proof that. 

Hint: you should be looking at $2^{3^{n+1}}$, not $2^{3^n+1}$. Assume, by induction that $2^{3^n} + 1 = 3^nd$, i.e., $2^{3^n} = 3^nd - 1$, and see what you can get from $2^{3^{n+1}} + 1 = 2^{3^n\cdot 3} + 1 = (2^{3^n})^3 + 1$.
