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In Denis Hanson's proof, he defines two terms:

(1) $B(n)$: which is the Least Common Multiple of $\{1,2,\dots,n\}$

$$B(n) = \prod\limits_{p^a \le n}p$$

(2) $C(n)$: an integer

$$C(n) = \dfrac{n!}{\lfloor n/a_1 \rfloor\lfloor n/a_2 \rfloor\lfloor n/a_3 \rfloor\dots}$$

where $a_1 = 2$ and $a_{n+1} = a_1 a_2 \dots a_n + 1$

The proof ends with the following conclusion:

$$C(n) < 3^n$$

I am completely lost where $C(n)$ was shown to be equal or greater to $B(n)$.

Am I misunderstanding?

If someone could explain how $C(n) \ge \text{LCM}(1,2,\dots,n)$, I would greatly appreciate it.

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Since $\alpha_p$ is such that $p^{\alpha_p}$ is the highest power of $p$ not exceeding $n$, we can write $$\alpha_p=[\log_pn]$$ So, $B(n)$ is written as $$B(n)=\prod_{p\le n}p^{[\log_pn]}$$

From LEMMA 2., we get $$[\log_pn]\le \beta_p(n)$$ from which $$B(n)=\prod_{p\le n}p^{[\log_pn]}\le \prod_{p\le n}p^{\beta_p(n)}=C(n)$$ follows.

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  • $\begingroup$ Thank you very much! This was greatly bothering me that I was not clear on this! $\endgroup$ – Larry Freeman Oct 10 '18 at 9:41

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