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 , if the conditions for n and m are respected.

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 powers 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.

I think that for n = 2, the conjecture B is more appropriate than the

Brocard's conjecture.

Brocard didn't take a risk in 1904 by saying that the first member of

the inequality 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 directly see that the description given by the

Brocard's conjecture does not follow the evolution of the number of

primes between the squares of consecutive primes.

In this case, it seems to me that he could propose m instead of 4.

Between the square of the 10000000000000000th prime and the square of

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 finally that my conjecture is more consistent and more

'descriptive'. But I am taking more risk, because my proposal can be

false ...

Questions :

1. Were these two conjectures already found and proposed before me ?

2. Can you prove these conjectures or give a counterexample ?

3. Can you propose a probabilistic or heuristic argument in favor of

the two conjectures ?

4. How are these conjectures related to other known theorems or

conjectures ?

5. Do you see some consequences of the formal similarities ?

6. Can you propose some interesting results about prime numbers,

supposing true these conjectures ?

7. Can you propose a conjecture giving a relation between Pi((m+1)^n),

Pi(m^n),Pi((p_(m+1))^n) and Pi((p_m)^n) ?

8. Can you improve the two conjectures and try to replace the

inequalities by 'equalities' ?

Regards,

Patrick Capelle.

Brussels.

patrick.capelle@...