Fermat Number whose Index is Sum of Integers

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Theorem

Let $F_n = 2^{\left({2^n}\right)} + 1$ be the $n$th Fermat number.

Let $k \in \Z_{>0}$.


Then:

$F_{n + k} - 1 = \left({F_n - 1}\right)^{2^k}$


Proof

By the definition of Fermat number

\(\displaystyle F_{n + k} - 1\) \(=\) \(\displaystyle 2^{2^{n + k} }\) $\quad$ Definition of Fermat Number $\quad$
\(\displaystyle \) \(=\) \(\displaystyle 2^{2^n 2^k}\) $\quad$ $\quad$
\(\displaystyle \) \(=\) \(\displaystyle \left({2^{2^n} }\right)^{2^k}\) $\quad$ $\quad$
\(\displaystyle \) \(=\) \(\displaystyle \left({F_n - 1}\right)^{2^k}\) $\quad$ Definition of Fermat Number $\quad$

$\blacksquare$


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