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Let $\text{lcm}(n)$ be the least common multiple of $(1, 2, \dots, \lfloor n\rfloor)$.

As I understand it, there is a well-known relationship between a factorial and the ratio of least common multiples (see my question here):

$$x! = \prod\limits_{i \ge 1}\text{lcm}\left(\frac{x}{i}\right)$$

For example:

$$6! = \text{lcm}(6)\text{lcm}(3)\text{lcm}(2) = (5\times4\times3)(3\times2)(2) = 720$$

Given:

$$\frac{(x^2+x)!}{(x^2)!} = \frac{\prod\limits_{i \ge 1}\text{lcm}((x^2+x)/i)}{\prod\limits_{j \ge 1}\text{lcm}((x^2)/j)}$$

Let $p$ be the highest prime less or equal to $x^2+x$.

Would it follow that there exists a combination of factorials such that:

$$\frac{(x^2+x)!\left\lfloor\frac{x^2}{2}\right\rfloor!\dots\left\lfloor\frac{x^2}{p}\right\rfloor!\left\lfloor\frac{x^2+x}{2\times3}\right\rfloor!\dots}{(x^2)!\left\lfloor\frac{x^2+x}{2}\right\rfloor!\dots\left\lfloor\frac{x^2+x}{p}\right\rfloor!\left\lfloor\frac{x^2}{2\times3}\right\rfloor!\dots} = \frac{\text{lcm}(x^2+x)}{\text{lcm}(x^2)}$$

The answers seems to me to be yes as long as all the combinations of primes $\{2, \dots, p\}$ are included in the correct way since this should cancel out the other $\text{lcm}$ values in the original equation.

Am I wrong?


Edit 1:

Here's an example with $x=4$:

$$\frac{20!\left(\frac{16}{2}\right)!\left\lfloor\frac{16}{3}\right\rfloor!\left\lfloor\frac{16}{5}\right\rfloor!\left\lfloor\frac{16}{7}\right\rfloor!\left\lfloor\frac{20}{6}\right\rfloor!\left(\frac{20}{10}\right)!}{16!\left(\frac{20}{2}\right)!\left\lfloor\frac{20}{3}\right\rfloor!\left(\frac{20}{5}\right)!\left\lfloor\frac{20}{7}\right\rfloor!\left\lfloor\frac{16}{6}\right\rfloor!\left\lfloor\frac{16}{10}\right\rfloor!}=\frac{\text{lcm}(4^2+4)}{\text{lcm}(4^2)} = 17\times19 = 323$$


Edit 2:

Mathlove pointed out a mistake in my original logic. I have changed the definition of $p$ to be the highest power less or equal to $x^2 + x$ (previously, it was $p$ to be the highest power less or equal to $x$ which was incorrect.

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  • $\begingroup$ For the last "equality", can you add an example for a small $x$, say $4$? It is difficult for me to understand what you want to say. $\endgroup$ – mathlove Oct 18 '18 at 4:52
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    $\begingroup$ Great suggestion. $x=4$ works. I've added it as an example to the question. $\endgroup$ – Larry Freeman Oct 18 '18 at 5:24
  • $\begingroup$ It looks you have $p=7$ for $x=4$, which seems to contradict "$p$ be the highest prime less or equal to $x$". $\endgroup$ – mathlove Oct 18 '18 at 5:52
  • $\begingroup$ Good point. That was a mistake. I should say that p is the highest power equal or less than $x^2+x$. Otherwise, it won't cancel out when $i$ gets higher than $x$ in the above equation. $\endgroup$ – Larry Freeman Oct 18 '18 at 5:54
  • $\begingroup$ "Let $p$ be the highest prime less or equal to $x^2+x$." Then, we have $p=19$ instead of $p=7$ for $x=4$, don't we? $\endgroup$ – mathlove Oct 18 '18 at 6:03

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