# Fermat Number whose Index is Sum of Integers

## 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 $\quad$

$\blacksquare$