Product of primes less than or equal to n

[R.P.F.]

Homework Statement

I'm working on a problem and as long as I can show that
$$\prod_{p\leq n}p \leq \prod_{n<p\leq 2n}p [/ tex]<br /> then I'm done. But I'm having trouble with this..Can someone help? <img src="https://cdn.jsdelivr.net/joypixels/assets/8.0/png/unicode/64/1f61b.png" class="smilie smilie--emoji" loading="lazy" width="64" height="64" alt=":-p" title="Stick Out Tongue :-p" data-smilie="7"data-shortname=":-p" /><br /> <h2>Homework Equations</h2><br /> <h2>The Attempt at a Solution</h2><br /> <br /> I tried to use PNT but could not solve it..<br /> EDIT: and maybe someone could also tell why my tex commands did not work out..?$$

 

[ArcanaNoir]

I can't really tell what you were going for with the tex. can you try to write it out with regular text and maybe then I can straighten out the tex?

 

[Petek]

Homework Statement

I'm working on a problem and as long as I can show that
$$\prod_{p\leq n}p \leq \prod_{n<p\leq 2n}p$$
then I'm done. But I'm having trouble with this..Can someone help?

Homework Equations

The Attempt at a Solution

I tried to use PNT but could not solve it..
EDIT: and maybe someone could also tell why my tex commands did not work out..?

I corrected your tex. There was an extra space between the final "/" and "tex".

 

[ArcanaNoir]

I corrected your tex. There was an extra space between the final "/" and "tex".

And I fail to see the obvious... :( I guess it's bed time. Really cool problem though! (not that I can solve it)

 

[ArcanaNoir]

So, it's not always true. Not true for n less than 8, and not true again for n=18 (510,510 and 392,863). I'm supposing it's not true for other values of n, but I think this is sufficient counter-example to rethink the situation.