<s>[INST] What is the greatest common divisor of $3^{3^{333}}+1$ and $3^{3^{334}}+1$? What is the greatest common divisor of $3^{3^{333}}+1$ and $3^{3^{334}}+1$ ? Could somebody please help me ?
 [/INST] Write $T_n=3^{3^n}+1$.  Then
$$\eqalign{T_n^3
  &=(3^{3^n})^3+3(3^{3^n})^2+3(3^{3^n})+1\cr
  &=3^{3^{n+1}}+1+3(3^{3^n})(3^{3^n}+1)\cr
  &=T_{n+1}+3^{3^n+1}T_n\ .\cr}$$
From this you should be able to see an impotant relationship between $T_n$ and $T_{n+1}$.
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