Exercise 1.26

Question

\def\imp{\Rightarrow}
\def\gcro{\mbox{[\hspace{-.15em}[}}
\def\dcro{\mbox{]\hspace{-.15em}]}}

ewcommand{\D}{\mathrm{d}}

ewcommand{\Q}{\mathbb{Q}}

ewcommand{\Z}{\mathbb{Z}}

ewcommand{\N}{\mathbb{N}}

ewcommand{\R}{\mathbb{R}}

ewcommand{\C}{\mathbb{C}}

ewcommand{\F}{\mathbb{F}}

ewcommand{e}{\,\mathrm{Re}\,}

ewcommand{\ord}{\mathrm{ord}}

ewcommand{
}{\mathrm{N}}

ewcommand{\legendre}[2]{\genfrac{(}{)}{}{}{#1}{#2}}

If $a^n + 1$ is a prime, show that $a$ is even and that $n$ is a power
of 2. Primes of the form $2^{2^t} + 1$ are called Fermat primes. For
example, $2^{2^1} + 1 = 5$ and $2^{2^2} + 1 = 17$.  It is not known if
there are infinitely many Fermat primes.

Richard Ganaye · Accepted Answer

\def\imp{\Rightarrow} \def\gcro{\mbox{[\hspace{-.15em}[}} \def\dcro{\mbox{]\hspace{-.15em}]}} ewcommand{\D}{\mathrm{d}} ewcommand{\Q}{\mathbb{Q}} ewcommand{\Z}{\mathbb{Z}} ewcommand{\N}{\mathbb{N}} ewcommand{\R}{\mathbb{R}} ewcommand{\C}{\mathbb{C}} ewcommand{\F}{\mathbb{F}} ewcommand{ e}{\,\mathrm{Re}\,} ewcommand{\ord}{\mathrm{ord}} ewcommand{ }{\mathrm{N}} ewcommand{\legendre}[2]{\genfrac{(}{)}{}{}{#1}{#2}} \begin{proof} If $a = 1, a^n + 1$ is a prime. Suppose $a >1$, and $n>1$. If $a$ was odd, $a^n+1>2$ is even, so is not a prime. Consequently, if $a^n+1$ is prime, $a>1$, then $a$ is even. Write $n = 2^t u$, where $u$ is odd. If $u>1$, then, from Ex. 24(b), we obtain $$a^n+1 = a^{2^tu}+1 = (a^{2^t} + 1)\sum_{i=0}^{u-1} (-1)^ia^{i2^t}.$$ So $11,n>1$), $a$ is even and $n$ is a power of $2$. \end{proof}

Exercise 1.26

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Exercise 1.26

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