Expand Messages
• ... What s the size of Pn#? (See http://primes.utm.edu/glossary/page.php?sort=Primorial ) Therefore what s the expected density of primes around an arbitrary
Message 1 of 2 , Mar 15, 2007
--- Wes & Paige Bullock <6bullocks@...> wrote:
> I found something interesting when I was investigating the density of primes
> near the large prime gaps around the primorials.
>
> Basically, it could be stated as the following conjecture:
>
> The number of primes less than a primorial value, Pn#, approaches P(n-1)#
> (in a percentage error sense) as n increases.

What's the size of Pn#? (See
http://primes.utm.edu/glossary/page.php?sort=Primorial )

Therefore what's the expected density of primes around an arbitrary number in
the vicinity of Pn#?

There's nothing special about primorials in this regard, they behave exactly as
arbitrary numbers of similar size.

Phil

() ASCII ribbon campaign () Hopeless ribbon campaign
/\ against HTML mail /\ against gratuitous bloodshed

[stolen with permission from Daniel B. Cristofani]

____________________________________________________________________________________
Be a PS3 game guru.
Get your game face on with the latest PS3 news and previews at Yahoo! Games.
http://videogames.yahoo.com/platform?platform=120121
• Proof: You don t need Li, PNT is enough. We start with a form of PNT: sum(ln p)(p lim(n- inf)(sum(ln p)(p lim(n- inf)(pn /
Message 2 of 2 , Mar 16, 2007
Proof:

You don't need Li, PNT is enough. We start with a form of PNT:
sum(ln p)(p<=pn) ~ pn. ==>
lim(n->inf)(sum(ln p)(p<=pn) / pn) = 1 ==>
lim(n->inf)(pn / sum(ln(p)(p<=pn) = 1 ==>
lim(n->inf)(pn / ln(pn#)(p<=pn) = 1 ==>
lim(n->inf)(pn*p(n-1)#) / (ln(pn#)*p(n-1)#) = 1 ==>
lim(n->inf)(pn#) / (ln(pn#)*p(n-1)#) ) = 1 ==>
lim(n->inf)(pn#) / ln(pn#) = p(n-1)# = PNT, qed.

Werner
Your message has been successfully submitted and would be delivered to recipients shortly.