Welcome to mathimatikoi.org forum; Enjoy your visit here.

Irrational number

Number theory
Post Reply
User avatar
Tolaso J Kos
Administration team
Administration team
Articles: 2
Posts: 861
Joined: Sat Nov 07, 2015 6:12 pm
Location: Larisa

Irrational number


Post by Tolaso J Kos » Tue Nov 10, 2015 4:02 pm

Let $N \in \mathbb{N} \mid N>1$. Prove that the number:

$$\mathcal{N}= \sqrt{1\cdot 2 \cdot 3 \cdot 4 \cdots (N-1)\cdot N}$$

is irrational.
Hidden message
May we have an alternate proof without Bertrand's postulate,or is this impossible?
Imagination is much more important than knowledge.
User avatar
Articles: 0
Posts: 174
Joined: Sat Nov 14, 2015 6:32 am
Location: Melbourne, Australia

Re: Irrational number


Post by Riemann » Wed Aug 01, 2018 9:33 am

Let $p \leq N$ be the last prime. If we prove that between $p$ and $N$ does not exist a number that has $p$ as a factor we are done. So, we need to prove that $2p>N$. But this is exactly what Bertrand's postulate says.
$\displaystyle \sum_{n=1}^{\infty}\frac{1}{n^s}= \prod_{p \; \text{prime}}\frac{1}{1-p^{-s}}$
Post Reply