Re: Equation for consecutive prime numbers
- --- In email@example.com,
Maximilian Hasler <maximilian.hasler@...> wrote:
> Letting p=q+d ...Yes, it seems to a very weak claim that the equation
> This is true whenever d^2/2 < p+q = 2q+d,
> for d=1 or any even value of d.
> No other condition on parity (let alone primality)
> of p,q seems required.
is satisfied for every pair of consecutive primes.
Yet there is no proof known to humankind!
We would like the gap d = p - q to be O(p^theta),
with theta < 1/2. That is not proven.
Not by Cramer, not by Hoheisel, not by
Montgomery, not by Heath-Brown.
Please see pp 253-254 of Ribenboim's book.
Even the assumption of the Riemann hypothesis
is not enough to prove the author's claim,
since that leads only to d = O(log(p)*sqrt(p)),
as far as any human knows.
Of course we strongly believe, like Cramer, that
d = O(log(p)^2). But belief ain't proof.