<s>[INST] Prove that the number $3^{3^n} + 1$ has at least $2n + 1$ prime factors. For any natural $n,$ prove that $3^{3^n} + 1$ has at least $2n + 1$ prime factors.
My idea was to use induction:


*

*for $n = 1$:
$$f(1) = 3^3 + 1 = 28 = 7*2^2$$

*let it be true for $n = k$, then for $n = k + 1$:
$$f(k + 1) = 3^{3^{k + 1}} + 1 = 3^{3*3^k} + 1 = (3^{3^k} + 1)(3^{2*3^k} - 3^{3^k} + 1) = f(k)\times(3^{2*3^k} - 3^{3^k} + 1)$$
Now I have a problem:   how to prove that $(3^{2*3^k} - 3^{3^k} + 1)$ is not a prime number?
Or, if it is harder than solving the original problem, please give a hint where I turned the wrong way.
 [/INST] $$\large(3^{2\times3^k} - 3^{3^k} + 1) = (3^{3^k}-3^{(3^k+1)/2}+1)(3^{3^k}+3^ {(3^k+1)/2}+1)$$
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