- --- In primenumbers@yahoogroups.com, "Bill" <bill2math@...> wrote:
>

http://tech.groups.yahoo.com/group/primenumbers/message/17020

> Aah! It's good to see my favorite math problem, the number of primes between squares, reappear on primenumbers, and while it is up, I would like to mention Capelle's wondeful conjecture A, posted here on primenumbers Sept 4, 2005, message number 16999. It is an extension of Legendre's conjecture, which he uses as a basis for his expansive formula.

> Capelle conjecture A is:

> Pi((m+1)^n) - Pi(m^n) >= m^(n-2) for n>=2, m>=1 .

> Legendre's conjecture claims there is at least one prime between the squares of consecutive integers, and is Capelle's case with n=2 and m = 1.

>

> I have some ideas how Capelle's conjecture can be made even more powerful, and I have done several handfulls of calculations to check the viability of my approach, but I of course need to do a lot more calculations. Can anyone direct me to a database that tells me the number of primes between two positive integers, or perhaps has someone checked out Capelle's conjecture A to any significant degree? Any leads or previous work would lighten my task.

>

> Thanks, Bill Oscarson

>

Patrick Capelle - --- In primenumbers@yahoogroups.com, "Bill" <bill2math@...> wrote:
>

Intuitively Legendre's conjecture would appear to be true: the

> Legendre's conjecture claims there is at least one prime between the squares of consecutive integers, and is Capelle's case with n=2 and m = 1.

>

rate of growth of the gap between squares looks like it would

far exceed the rate of growth in the average gap between primes.

However the pattern of these gaps can be very uneven and I believe that

it HAS been proven that the gap between primes can be arbitrarily

large. This does not disprove Legendre's conjecture - probably

the only way to do that would be to find a counterexample. Good

with that project!

a. - On Sun, 2010-12-12 at 10:13 +0000, Aldrich wrote:
> However the pattern of these gaps can be very uneven and I believe

The proof is trivial. For prime p, all integers n s.t.

> that

> it HAS been proven that the gap between primes can be arbitrarily

> large. This does not disprove Legendre's conjecture - probably

p!+1 < n <= p!+p are divisible by at least one prime <= p and p can be

taken arbitrarily large.

Paul