Exercise 2.8

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}}

Use Exercise 7 to show that there are infinitely many 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 the set $\mathbb{P}$ of prime numbers was finite, we obtain from Ex.2.7, for all $n \geq 2$,
$$\sqrt[n]{n!} \leq C = \prod_{p\in \mathbb{P}} p^{\frac{1}{p-1}},$$
where $C$ is an absolute constant.

Yet  $\lim\limits_{n	o \infty} \sqrt[n]{n!}  = + \infty$. Indeed
\begin{align*}
\ln(\sqrt[n]{n!}) &= \frac{1}{n}(\ln 1 + \ln 2 + \cdots + \ln n)\
\end{align*}
As $\ln$ is an increasing fonction, 
\begin{align*}
\int_{i-1}^i \ln t\, \D t &\leq \ln i, \ i=2,3,\ldots,n
\end{align*}
So
\begin{align*}
\int_1^n \ln t \, \D t &= \sum_{i=2}^n \int_{i-1}^i \ln t \, \D t \leq \sum_{i=2}^n \ln i =  \sum_{i=1}^n \ln i
\end{align*}
Thus
\begin{align*}
\ln(\sqrt[n]{n!}) &\geq  \frac{1}{n} \int_1^n \ln t\,  \D t\ = \frac{1}{n}(n \ln n - n +1) = \ln n - 1 + \frac{1}{n}
\end{align*}
As $\lim\limits_{n	o \infty}  \ln n - 1 + \frac{1}{n} = + \infty$, $\lim\limits_{n	o \infty} \ln(\sqrt[n]{n!}) = + \infty$, so $\lim\limits_{n	o \infty} \sqrt[n]{n!} = + \infty$.

Thus there exists $n$ such that $\sqrt[n]{n!} \geq C$: this is a contradiction. $\mathbb{P}$ is an infinite set.
\end{proof}

Exercise 2.8

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Exercise 2.8

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