Help Needed

[Richard Guy]

R.~C.~Baker, Glyn Harman \& J.~Pintz, The difference
between consecutive primes, II, {\it Proc.\ London
Math.\ Soc.}(3) {\bf83}(2001) 522--562; {\it MR}
{\bf2002f}:11125.

improve theta to 0.525.  Does anyone know of a
further improvement ?       R.

On Mon, 1 Jan 2007, Jonathan Post wrote:

> Once again, a famous classic problem stumbled upon, and 
> a trivial small part
> of the range found encouraging.  Not even new to OEIS. 
> Much harder than it
> seems, having been wrtestled with by major 
> mathematicians for centuries.
>
> That's Legendre's
> conjecture<http://mathworld.wolfram.com/LegendresConjecture.html>that
> for every [image:
> n] there exists a prime 
> <http://mathworld.wolfram.com/PrimeNumber.html> [image:
> p] between [image: n^2] and [image: (n+1)^2] (Hardy and 
> Wright 1979, p. 415;
> Ribenboim 1996, pp. 397-398).
>
> It is the 3rd of the 4 Landau Problems.  See:
>
> Weisstein, Eric W.
> <http://mathworld.wolfram.com/about/author.html>"Landau's 
> Problems."
> From
> *MathWorld* <http://mathworld.wolfram.com/>--A Wolfram 
> Web Resource.
> http://mathworld.wolfram.com/LandausProblems.html
> Although it is not known if there always exists a
> prime<http://mathworld.wolfram.com/PrimeNumber.html> 
> [image:
> p] between [image: n^2] and [image: (n+1)^2], Chen 
> (1975) has shown that a
> number [image: P] which is either a
> prime<http://mathworld.wolfram.com/PrimeNumber.html>or
> semiprime <http://mathworld.wolfram.com/Semiprime.html> 
> does always satisfy
> this inequality. Moreover, there is always a prime 
> between [image:
> n-n^theta] and [image: n] where [image: theta==23/42] 
> (Iwaniec and Pintz
> 1984; Hardy and Wright 1979, p. 415). The smallest
> primes<http://mathworld.wolfram.com/PrimeNumber.html>between 
> [image:
> n^2] and [image: (n+1)^2] for [image: n==1], 2, ..., are 
> 2, 5, 11, 17, 29,
> 37, 53, 67, 83, ... (Sloane's
> A007491<http://www.research.att.com/%7Enjas/sequences/A007491>
> ).
>
> On 1/1/07, Artur <grafix at csl.pl> wrote:
>> 
>> Dear Seqfans,
>> This picture not confirmed, not proofed yet hypothesis 
>> that between
>> squares of two successive positive numbers occured 1 
>> or more prime. I'm
>> not happy yet from approximation curve (magenta)
>> 
>> Surely these will be between squares of two successive 
>> positive numbers
>> occured 2 or more primes
>> in general but these number increased with n and I 
>> can't imagine that for
>> two very big numbers will be go down to 1
>> 
>> BEST WISHES
>> ARTUR
>> 
>> 
>> 
>