Exercise 1.30

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}}

Prove that  $H_n = \frac{1}{2} + \frac{1}{3} + \ldots + \frac{1}{n}$
is not an integer.

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} Let $s$ such that $2^{s} \leq n < 2^{s+1}$ ($s = \left \lfloor \frac{\ln n}{\ln 2} ight floor \geq 1$). $$H_n = \frac{1}{2} + \cdots + \frac{1}{n} = \frac{\sum_{i=2}^n a_i}{n!}, \qquad \mathrm{where}\ a_i = \frac{n!}{i} \in \Z.$$ Let $k = \ord_2(n!)$. We will show that $\ord_s(a_i)$ is minimal for $i_0 = 2^s$, where $\ord_2(a_{i_0}) = k - s$, and that this minimum is reached only for this index $i_0$. Indeed, each $i$ such that $2 \leq i \leq n$ can be written with the form $i = 2^tq, 2 mid q$. Then $i = 2^t q \leq n < 2^{s+1}$, so $2^t < 2^{s+1}, t < s+1, t \leq s$, which proves $$\ord_2(a_i) = k - t \geq k - s = \ord_2(a_{i_0}).$$ Moreover, if $\ord_2(a_i) = \ord_2(a_{i_0})$, then $k - t = k - s$, so $s = t$. Since $s=t$, $i = 2^s q$, where $2 mid q$. If $q>1$, then $i\geq 2^{s+1}>n$ : it's impossible. So $q=1$ and $i = 2^s = i_0$. Using Ex 1.21, we see that $$\ord_2\left(\sum_{i=2}^n a_i ight)= \ord_2(a_{i_0}) = k - s < k = \ord_2(n!).$$ So $$H_n = \frac{2^{k-s} Q}{2^k R} = \frac{Q}{2^s R},$$ where $Q,R$ are odd integers. $H_n$ is a quotient of an odd integer by an even integer: $H_n$ is never an integer. \end{proof}

Exercise 1.30

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Exercise 1.30

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