--- In

primenumbers@yahoogroups.com,

Maximilian Hasler <maximilian.hasler@...> wrote:

> Letting p=q+d ...

> 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.

Yes, it seems to a very weak claim that the equation

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.

David