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| | The author is [http://www.myelectronicmd.com/get_reference.php?Id=1296&condition=ANAL%20HERPES%20SIMPLEX%20VIRUS&symname=A&typ=3 called Irwin] Wunder but it's not the most masucline title out there. Puerto Rico is exactly where he's always [http://ghaziabadmart.com/oxwall/blogs/post/5933 std testing at home] [http://wmazowiecku.pl/stay-yeast-infection-free-using-these-helpful-suggestions/ wmazowiecku.pl] been residing but she needs to move because of her family members. Since she was 18 she's been operating as a meter reader but she's always wanted her personal company. The [http://Men.Webmd.com/features/6-most-common-std-men favorite] hobby for my at home at home std testing std test children and me is to perform baseball but I haven't made a dime with it.<br><br>Also visit my website - [http://www.1a-pornotube.com/blog/84958 1a-pornotube.com] |
| '''Legendre's conjecture''', proposed by [[Adrien-Marie Legendre]], states that there is a [[prime number]] between ''n''<sup>2</sup> and (''n'' + 1)<sup>2</sup> for every [[positive integer]] ''n''. The [[conjecture]] is one of [[Landau's problems]] (1912) and remains unsolved.
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| [[File:Plot of number of primes between consecutive squares.png|thumbnail|right|Plot of the number of primes between n² and (n+1)² {{OEIS|A014085}}]] | |
| The [[prime number theorem]] suggests the actual number of primes between ''n''<sup>2</sup> and (''n'' + 1)<sup>2</sup> {{OEIS|A014085}} is about ''n''/ln(''n''), i.e. about as many as the [[Prime-counting function|number of primes less than or equal to ''n'']].
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| If Legendre's conjecture is true, the [[prime gap|gap]] between any two successive primes would be <math>O(\sqrt p)</math>. In fact the conjecture follows from [[Andrica's conjecture]] and from [[Oppermann's conjecture]]. [[Harald Cramér]] [[Cramér's conjecture|conjectured]] that the gap is always much smaller, <math>O(\log^2 p)</math>; if Cramér's conjecture is true, Legendre's conjecture would follow for all sufficiently large numbers. Cramér also proved that the [[Riemann hypothesis]] implies a weaker bound of <math>O(\sqrt p\log p)</math> on the size of the largest prime gaps. Legendre's conjecture implies that at least one prime can be found in every revolution of the [[Ulam spiral]].
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| Because the conjecture follows from Andrica's conjecture, it suffices to check that each prime gap starting at ''p'' is smaller than <math>2\sqrt p.</math> A table of maximal [[prime gaps]] shows that the conjecture holds to 10<sup>18</sup>. A counterexample near 10<sup>18</sup> would require a prime gap fifty million times the size of the average gap.
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| ==See also==
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| * [[Bertrand's postulate]]
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| * [[Brocard's conjecture]]
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| * [[Firoozbakht’s conjecture]]
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| ==External links==
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| * {{mathworld|urlname=LegendresConjecture|title=Legendre's conjecture}}
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| [[Category:Conjectures about prime numbers]]
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| [[Category:Unsolved problems in mathematics]]
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| {{Numtheory-stub}}
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Latest revision as of 06:04, 13 December 2014
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