Is my solution correct?

Prove that :

\[1 + \frac{1}{2} + \frac{1}{3} + .... + \frac{1}{n} \notin \mathbb{Z} (\forall n \in \mathbb{Z}) \]

Prove :

Assume $1 + \frac{1}{2} + \frac{1}{3} + .... + \frac{1}{n} \notin \mathbb{Z}$

So that : $1 + \frac{1}{2} + \frac{1}{3} + .... + \frac{1}{n} = \frac{a}{b} ; (a,b) = 1$

We must demonstrate : $1 + \frac{1}{2} + \frac{1}{3} + .... + \frac{1}{n+1} \notin \mathbb{Z}$

Pretend that : $1 + \frac{1}{2} + \frac{1}{3} + .... + \frac{1}{n+1} \in \mathbb{Z}$

$\Rightarrow \frac{a}{b} + \frac{1}{n + 1} \in \mathbb{Z}$ $\Rightarrow \frac{a(n + 1) + b}{b(n + 1)} \in \mathbb{Z}$ $\Rightarrow a(n + 1) + b \vdots b(n + 1)$

By (a,b) = 1 then $n + 1 \vdots b$ and $b \vdots n + 1$ so $n+1 = b$

But :

$1 + \frac{1}{2} + \frac{1}{3} + .... + \frac{1}{n} = \frac{a}{n+1} = \frac{c}{n!}$ Thus, $n! \vdots n+1 ;(absurdity)$

Therefore : $1 + \frac{1}{2} + \frac{1}{3} + .... + \frac{1}{n} \notin \mathbb{Z} (\forall n \in \mathbb{Z})$

Thanks in advance ~!!

#NumberTheory

Easy Math Editor

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Comments

Good attempt. However, there is an error in the last line. So what if we have $\frac{a}{ n+1} = \frac{c}{n!}$ ? Why does that mean that $n! \mid n+1$ ? For example, $\frac{2}{4} = \frac{ 3}{6}$ , and 4 and 6 do not divide each other.

If instead you want to argue that $n + 1 \not \mid n!$ , then consider $n = 5$ .

Calvin Lin Staff - 5 years, 10 months ago

Thanks sir, but am I in the right track to moving on or i should change my methods ? Thanks in advance !

Rony Phong - 5 years, 10 months ago

You are most likely on the wrong track, in that I believe there is no solution arising from your current approach.

See Pi Han's comment stream for suggestions on how to solve this problem.

Calvin Lin Staff - 5 years, 10 months ago

A simpler approach is to apply Bertrand's Postulate.

Pi Han Goh - 5 years, 10 months ago

That is an overkill for this problem. There is a much simpler approach.

Calvin Lin Staff - 5 years, 10 months ago

Yes. It involves the highest power of 2.

Pi Han Goh - 5 years, 10 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$