Product of primes less than or equal to n

  • Thread starter Thread starter R.P.F.
  • Start date Start date
  • Tags Tags
    Primes Product
R.P.F.
Messages
210
Reaction score
0

Homework Statement



I'm working on a problem and as long as I can show that
\prod_{p\leq n}p \leq \prod_{n&lt;p\leq 2n}p [/ tex]<br /> then I&#039;m done. But I&#039;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..?
 
Physics news on Phys.org
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?
 
R.P.F. said:

Homework Statement



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

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".
 
Petek said:
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)
 
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.
 
Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...
Back
Top