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.<br>
<br>
That's <a href="http://mathworld.wolfram.com/LegendresConjecture.html" class="Hyperlink">Legendre's conjecture</a> that for every <img src="http://mathworld.wolfram.com/images/equations/LandausProblems/inline1.gif" class="inlineformula" alt="n" border="0" height="15" width="7">
 there exists a <a href="http://mathworld.wolfram.com/PrimeNumber.html" class="Hyperlink">prime</a> <img src="http://mathworld.wolfram.com/images/equations/LandausProblems/inline2.gif" class="inlineformula" alt="p" border="0" height="15" width="9">
 between <img src="http://mathworld.wolfram.com/images/equations/LandausProblems/inline3.gif" class="inlineformula" alt="n^2" border="0" height="16" width="14"> and <img src="http://mathworld.wolfram.com/images/equations/LandausProblems/inline4.gif" class="inlineformula" alt="(n+1)^2" border="0" height="16" width="43">
 (Hardy and
 Wright 1979, p. 415; Ribenboim 1996, pp. 397-398).<br>
<br>
It is the 3rd of the 4 Landau Problems.  See:<br>
<p class="citation">
<a href="http://mathworld.wolfram.com/about/author.html">Weisstein, Eric W.</a> "Landau's Problems."
 From <a href="http://mathworld.wolfram.com/"><i>MathWorld</i></a>--A Wolfram Web Resource. <a href="http://mathworld.wolfram.com/LandausProblems.html">http://mathworld.wolfram.com/LandausProblems.html</a>
</p>

Although it is not known if there always exists a <a href="http://mathworld.wolfram.com/PrimeNumber.html" class="Hyperlink">prime</a> <img src="http://mathworld.wolfram.com/images/equations/LandausProblems/inline7.gif" class="inlineformula" alt="p" border="0" height="15" width="9">
 between <img src="http://mathworld.wolfram.com/images/equations/LandausProblems/inline8.gif" class="inlineformula" alt="n^2" border="0" height="16" width="14"> and <img src="http://mathworld.wolfram.com/images/equations/LandausProblems/inline9.gif" class="inlineformula" alt="(n+1)^2" border="0" height="16" width="43">
, Chen (1975)
 has shown that a number <img src="http://mathworld.wolfram.com/images/equations/LandausProblems/inline10.gif" class="inlineformula" alt="P" border="0" height="15" width="9"> which is either
 a <a href="http://mathworld.wolfram.com/PrimeNumber.html" class="Hyperlink">prime</a> or <a href="http://mathworld.wolfram.com/Semiprime.html" class="Hyperlink">semiprime</a> does always satisfy this inequality. Moreover, there
 is always a prime between <img src="http://mathworld.wolfram.com/images/equations/LandausProblems/inline11.gif" class="inlineformula" alt="n-n^theta" border="0" height="16" width="35"> and <img src="http://mathworld.wolfram.com/images/equations/LandausProblems/inline12.gif" class="inlineformula" alt="n" border="0" height="15" width="7">
 where <img src="http://mathworld.wolfram.com/images/equations/LandausProblems/inline13.gif" class="inlineformula" alt="theta==23/42" border="0" height="15" width="62"> (Iwaniec
 and Pintz 1984; Hardy and Wright 1979, p. 415). The smallest <a href="http://mathworld.wolfram.com/PrimeNumber.html" class="Hyperlink">primes</a> between <img src="http://mathworld.wolfram.com/images/equations/LandausProblems/inline14.gif" class="inlineformula" alt="n^2" border="0" height="16" width="14">
 and <img src="http://mathworld.wolfram.com/images/equations/LandausProblems/inline15.gif" class="inlineformula" alt="(n+1)^2" border="0" height="16" width="43"> for <img src="http://mathworld.wolfram.com/images/equations/LandausProblems/inline16.gif" class="inlineformula" alt="n==1" border="0" height="15" width="33">
, 2, ..., are 2, 5, 11, 17, 29, 37, 53, 67, 83,
 ... (Sloane's <a href="http://www.research.att.com/%7Enjas/sequences/A007491" class="Hyperlink">A007491</a>).<br><br><div><span class="gmail_quote">On 1/1/07, <b class="gmail_sendername">Artur</b> <<a href="mailto:grafix@csl.pl">
grafix@csl.pl</a>> wrote:</span><blockquote class="gmail_quote" style="border-left: 1px solid rgb(204, 204, 204); margin: 0pt 0pt 0pt 0.8ex; padding-left: 1ex;">Dear Seqfans,<br>This picture not confirmed, not proofed yet hypothesis that between
<br>squares of two successive positive numbers occured 1 or more prime. I'm<br>not happy yet from approximation curve (magenta)<br><br>Surely these will be between squares of two successive positive numbers<br>occured 2 or more primes
<br>in general but these number increased with n and I can't imagine that for<br>two very big numbers will be go down to 1<br><br>BEST WISHES<br>ARTUR<br><br><br></blockquote></div><br>