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
\(\ds F_{n + k} - 1\) | \(=\) | \(\ds 2^{2^{n + k} }\) | 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 $9$