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

\(\ds F_{n + k} - 1\)

\(=\)

\(\ds 2^{2^{n + k} }\)

Definition of Fermat Number

\(\ds \)

\(=\)

\(\ds 2^{2^n 2^k}\)

\(\ds \)

\(=\)

\(\ds \left({2^{2^n} }\right)^{2^k}\)

\(\ds \)

\(=\)

\(\ds \left({F_n - 1}\right)^{2^k}\)

Definition of Fermat Number

$\blacksquare$

Sources

• 1982: P.M. Cohn: Algebra Volume 1 (2nd ed.) ... (previous) ... (next): $\S 2.4$: The rational numbers and some finite fields: Further Exercises $9$