Re: Two new conjectures about primes


Patrick Capelle wrote:
> I would like to propose two new conjectures about the quantity of primes in a given range.
> Date of discovery:15 August 2005 (if I am the first ...).
> They are presented at the same time because they have formal similarities.
>
> Conjecture A :
> pi((m+1)^n) - pi(m^n) >= m^(n-2)
> for n>= 2, m >=1.
> It means that there are at least m^(n-2) primes between m^n and (m+1)^n.
>
(snip)

> 2. Can you propose a probabilistic or heuristic argument in favour of the two conjectures ?
(snip)

You are dealing with a massively researched topic, so it is unlikely
that no one has thought of things along those lines. Have you done a
computer search for counterexamples?

Conjecture A is at least plausible given the prime number theorem (~
stands for approximately):

pi((m+1)^n) - pi(m^n) ~ [(m+1)^n]/(n*ln(m+1)) - [m^n]/(n*ln(m))
~ [(m+1)^n - m^n] /(n*ln(m))
= [(m^n + n*m^(n-1) + lower order terms) -
m^n]/(n*ln(m))
>= [n*m^(n-1)]/(n*ln(m)]
= [m^(n-1)]/ln(m) >= m^(n-1) / m = m^(n-2).

You clould try something like that for conjecture B

Hope that helps

John Coleman

.

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