Help Needed

[Jonathan Post]

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
>
>
>
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