<s>[INST] 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. 
 [/INST] Hint: Use the fact that $$(2^{3^n})^3 + 1 = \left(2^{3^n} +1\right) \left((2^{3^n})^2- 2^{3^n} +  1\right)$$
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