Two new conjectures about primes

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.

Conjecture B :
pi((p_(m+1))^n) - pi((p_m)^n) >= m^(n-1)
for n>=1, m>=1.
It means that there are at least m^(n-1) primes between the nth power of two consecutive primes.

These two conjectures are inspired respectively by those of Legendre and Brocard :

1. The Legendre's conjecture :
There exists a prime p between m^2 and (m+1)^2 for every integer m [Hardy and Wright 1979, p.415 ; Ribenboim 1996, pp. 397-398].
Note that this conjecture is a particular case of the conjecture A (see n = 2).

2. The Brocard's conjecture :
pi((p_(m+1))^2) - pi((p_m)^2) >= 4 for m >= 2, which means that there are at least four primes between the squares of two consecutive primes greater than two. The conjecture B is more appropriate than the Brocard's conjecture. Brocard didn't take a risk in 1904 by saying that between the squares of two consecutive primes greater than two the number of primes is greater than a (little) constant. If you look at the drawing on http://mathworld.wolfram.com/BrocardsConjecture.html you will immediately understand what I mean : the values on average are increasing, and you can rapidly guess that the description given by the Brocard's conjecture does not follow the evolution of the number of primes between squares of consecutive primes.
Between the squares of the 10000000000000000th prime and the 10000000000000001th prime I am sure that there are at least 4 primes, but it's better now to ask the following relevant question : are there at least 10000000000000000 primes between them ?
I think that my conjecture is more consistent and more 'descriptive'. But I take more risk and my proposal can be false ...


Questions :

1. Can you prove these conjectures or give a counterexample ?
2. Can you propose a probabilistic or heuristic argument in favour of the two conjectures ?
3. How are these conjectures related to other known conjectures or theorems ?
4. Do you see some consequences of the formal similarities of the two conjectures ?
5. Can you say something about the relations between p((m+1)^n), p( m^n), (p_(m+1))^n and (p_m)^n ?
6. Can you propose some interesting results about prime numbers, supposing true these conjectures ?

Regards,
Patrick Capelle.
.

Relevant Pages

  • Re: A simple problem of number theory
    ... There is no counterexample if s - 1 = P.Q for distinct primes ... I'm not suggesting that condition is necessary. ... we can immediately conclude that the only "p and s-p spanning" ...
    (sci.math)
  • Re: A simple problem of number theory
    ... There is no counterexample if s - 1 = P.Q for distinct primes ... I'm not suggesting that condition is necessary. ... we can immediately conclude that the only "p and s-p spanning" ...
    (sci.math)
  • Re: goldbachs conjecture
    ... Curiously according to this, are just the primes, ... Whether or not Goldbach's conjecture is true or not, ... Popper called the Goldbach Conjecture true if, ... when it comes on a counterexample. ...
    (sci.math)
  • Re: Goldbach minus
    ... about any even number being describable as the sum of two primes. ... Perhaps a counterexample is known. ... counterexample known to the stronger conjecture that there are an infinite ... conjecture that there are an infinite of such adjacent-prime pairs for every ...
    (sci.math)
  • Re: Who thinks Goldbachs Conjecture is unprovable?
    ... > Goldbach Conjecture is a probabilistic assertion disguised in an analytic ... it's very improbable that suming all the pairs of primes ... The search of a counterexample ought to be discarded. ...
    (sci.math)