Re: A New Conjecture?
- --- In firstname.lastname@example.org,
"ronaldpeterdwyer" <dwyer_ron@...> conjectured:
> For any positive integer N>3, there is a prime numberNoting the counterexamples for N = 125 and N = 126
> between N and N-(sqrt N).
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.]