--- In

primenumbers@yahoogroups.com,

"ronaldpeterdwyer" <dwyer_ron@...> conjectured:

> For any positive integer N>3, there is a prime number

> between N and N-(sqrt N).

Noting the counterexamples for N = 125 and N = 126

that Jack has given, I remark on the more general

problem of proving that, for some fixed theta,

there is always a prime between (N - N^theta) and N,

for sufficiently large N (depending on theta).

There is a proof for theta = 21/40 :

R. C. Baker, G. Harman, and J. Pintz,

The difference between consecutive primes, II,

Proc. London Math. Soc., 83 (2001), 532-562.

If one assumes the Riemann hypothesis,

then a similar result follows for theta > 1/2.

I know of no such argument for theta = 1/2.

It has, of course, long been *conjectured*

that such a result obtains for any theta > 0.

[Ribenboim attributes this conjecture to Piltz, in 1884.]

David