<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] $$2^{3^m}+1=h\cdot 3^{m}\implies 2^{3^{m+1}}+1=(h\cdot 3^m-1)^3+1=\\=(h\cdot 3^m)((h\cdot 3^{m}-1)^2-(h\cdot 3^m-1)+1)=\\=(h\cdot 3^m)(h^2\cdot 3^{2m}-3h\cdot 3^m+3)$$ if  $m\ge 1$, then $3$ divides $h^2\cdot3^{2m}-3h\cdot 3^m+3$
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