35.10.222.135: Corrected Henrickson->Hendrickson

2013-07-16T20:07:44Z

Corrected Henrickson->Hendrickson

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[[File:Wikipedia primegaps.png|thumb|294px|Prime gap function ]]
In [[number theory]], '''Firoozbakht’s conjecture''', also known as the Firoozbakht conjecture,<ref name="Ribenboim">{{harvnb|Paulo Ribenboim|2004|p=185}}</ref><ref>{{cite web|last=Rivera|first=Carlos|title=Conjecture 30. The Firoozbakht Conjecture|url=http://www.primepuzzles.net/conjectures/conj_030.htm|accessdate=22 August 2012}}</ref><ref>{{cite web|last=Sinha|first=Nilotpal Kanti|title=On a new property of primes that leads to a generalization of Cramer's conjecture|url=http://arxiv.org/abs/1010.1399v2|publisher=preprint|accessdate=22 August 2012}}</ref> states that <math>p_{n}^{1/n}\, </math> (where <math>p_n\, </math> is the ''n''th prime) is a strictly decreasing function of ''n'', i.e.,

:<math>p_{n}^{1/n} > p_{n+1}^{1/(n+1)}</math> for all <math> n \ge 1.</math>

Equivalently: <math>p_{n+1} < p_{n}^{1+1/n}</math> for all <math> n \ge 1.</math>

Another equivalent statement, <math>\left(\frac{p_{n+1}}{p_{n}}\right)^n < p_{n}</math> for all <math> n \ge 1,</math> is the motivating reason for the sequence {{OEIS2C|id=A205827}}.

The conjecture is named after Farideh Firoozbakht, from the [[University of Isfahan]], who stated it in 1982. If this conjecture is true, then the [[prime gap]] function <math>g_n = p_{(n+1)} - p_{n} </math> satisfies <math> g_n < (\log p_{n})^2 - \log p_{n} </math> which is sharper than [[Cramér's conjecture]] <math> g_n = O((\log p_n)^2).</math> Currently, the largest stated verification was done by using "[[Prime gaps#Numerical results|Maximal gaps]] between consecutive primes less than 4.444 * 10<sup>12</sup>" by Firoozbakht. Using a table of maximal gaps and the above gap inequality, the confirmation value can be extended to all primes below 4{{e|18}}.<ref>[http://www.ieeta.pt/~tos/gaps.html Gaps between consecutive primes]</ref>

The conjecture is believed to be false, as it contradicts the [[Cramér's conjecture#Cramér–Granville conjecture|Cramér–Granville heuristic]]. If it were true it would imply the weaker [[Cramér conjecture]].

==See also==
*[[Prime number theorem]]
*[[Andrica's conjecture]]
* [[Legendre's conjecture]]
* [[Oppermann's conjecture]]

==Notes==
{{Reflist|colwidth=80em}}

==References==

* {{cite book|last=Ribenboim|first=Paulo|title=The Little Book of Bigger Primes Second Edition | publisher=Springer-Verlag
|year=2004|isbn=0-387-20169-6|ref=harv }}

{{Reflist}}

{{Prime number classes}}

{{DEFAULTSORT:Firoozbakht's conjecture}}
[[Category:Conjectures about prime numbers]]

Rigidity matroid - Revision history

35.10.222.135: Corrected Henrickson->Hendrickson