查看“︁菲鲁兹巴赫特猜想”︁的源代码

[[File:Wikipedia primegaps.png|thumb|450px|質數間隙函數]]

在[[數論]]中，'''菲鲁兹巴赫特猜想'''（Firoozbakht's conjecture 或 Firoozbakht conjecture<ref name="Ribenboim">{{cite book|last=Ribenboim|first=Paulo|title=The Little Book of Bigger Primes Second Edition |year=2004|url=https://archive.org/details/littlebookbigger00ribe_610|url-access=limited| publisher=Springer-Verlag| page=[https://archive.org/details/littlebookbigger00ribe_610/page/n209 185]|isbn=9780387201696 }}</ref><ref name=ppagc30>{{cite web|last=Rivera|first=Carlos|title=Conjecture 30. The Firoozbakht Conjecture|url=http://www.primepuzzles.net/conjectures/conj_030.htm|access-date=22 August 2012|archive-date=2016-03-03|archive-url=https://web.archive.org/web/20160303232238/http://www.primepuzzles.net/conjectures/conj_030.htm|dead-url=no}}</ref>）是數學上關於質數分布的一個猜想。該猜想以伊朗數學家{{link-en|法丽德·菲鲁兹巴赫特|Farideh Firoozbakht}}的名字命名，她於1982年提出此猜想。

該猜想聲稱，<math>p_{n}^{1/n}</math>是一個嚴格遞減函數（其中<math>p_n</math>是第<math>n</math>個質數），也就是說

:<math>\sqrt[n+1]{p_{n+1}} < \sqrt[n]{p_n} \qquad \text{ for all } n \ge 1.</math>

或等價地

:<math>p_{n+1} < p_n^{1+\frac{1}{n}} \qquad \text{ for all } n \ge 1,</math>

相關內容可見{{OEIS2C|id=A182134}}及{{OEIS2C|id=A246782}}。

藉由使用[[質數間隙|最大質數間隙]]（maximal gap）表，法丽德·菲鲁兹巴赫特確認她的猜想對大到<math>4.444\times 10^{12}</math>的數都成立。<ref name=ppagc30/>利用廣度更大的最大質數間隙表，目前已知該猜想對任何小於<math>2^{64}\sim1.84\times 10^{19}</math>的質數都成立。<ref>{{Cite web |url=http://www.ieeta.pt/~tos/gaps.html |title=Gaps between consecutive primes |access-date=2024-01-09 |archive-date=2012-09-10 |archive-url=https://web.archive.org/web/20120910134123/http://www.ieeta.pt/~tos/gaps.html |dead-url=no }}</ref><ref name="Kourbatov2018">{{cite web|last=Kourbatov|first=Alexei|title=Prime Gaps: Firoozbakht Conjecture|url=http://www.javascripter.net/math/primes/firoozbakhtconjecture.htm|access-date=2024-01-09|archive-date=2017-03-22|archive-url=https://web.archive.org/web/20170322234500/http://www.javascripter.net/math/primes/firoozbakhtconjecture.htm|dead-url=no}}</ref>

若此猜想成立，那麼[[質數間隙]]函數<math>g_n = p_{n+1} - p_n </math>會滿足下列關係：<ref>{{cite arXiv |last=Sinha |first=Nilotpal Kanti |title=On a new property of primes that leads to a generalization of Cramer's conjecture |year=2010 |pages= |class=math.NT |eprint=1010.1399|mode=cs2}}.</ref>

:<math> g_n < (\log p_n)^2 - \log p_n \qquad \text{ for all } n > 4.</math>

此外，<ref>{{citation |last=Kourbatov |first=Alexei |title=Upper bounds for prime gaps related to Firoozbakht's conjecture |journal=Journal of Integer Sequences |arxiv=1506.03042 |year=2015 |volume=18 |issue=Article 15.11.2 |url=http://cs.uwaterloo.ca/journals/JIS/VOL18/Kourbatov/kourb7.html |bibcode= |mr=3436186 |zbl=1390.11105 |accessdate=2024-01-09 |archive-date=2016-06-04 |archive-url=https://web.archive.org/web/20160604102023/https://cs.uwaterloo.ca/journals/JIS/VOL18/Kourbatov/kourb7.html |dead-url=no }}.</ref>

:<math> g_n < (\log p_n)^2 - \log p_n - 1 \qquad \text{ for all } n > 9,</math>

對此可見{{OEIS2C|id=A111943}}。

該猜想是對質數間隙上界最強的猜想之一，甚至比[[克拉梅爾猜想]]和尚克斯猜想（Shanks' Conjecture）還強。<ref name="Kourbatov2018" />從該猜想可推出強克拉梅爾猜想，而這與{{link-en|安德鲁·格兰维尔|Andrew Granville}}、{{link-hu|平茨·亚诺什|Pintz János}}<ref>{{citation |last=Granville |first=A. |author-link=Andrew Granville |title=Harald Cramér and the distribution of prime numbers | journal = Scandinavian Actuarial Journal |volume=1 |year=1995 |pages=12–28 |doi=10.1080/03461238.1995.10413946 |mr=1349149 |zbl=0833.01018 | url=http://www.dartmouth.edu/~chance/chance_news/for_chance_news/Riemann/cramer.pdf |archive-url=https://web.archive.org/web/20160502155118/http://www.dartmouth.edu/~chance/chance_news/for_chance_news/Riemann/cramer.pdf |archive-date=2016-05-02 }}.</ref><ref>{{citation |last=Granville |first=Andrew |author-link=Andrew Granville |title=Unexpected irregularities in the distribution of prime numbers |journal=Proceedings of the International Congress of Mathematicians |volume=1 |year=1995 |pages=388–399 |doi=10.1007/978-3-0348-9078-6_32 |isbn=978-3-0348-9897-3 |zbl=0843.11043 |url=http://www.dms.umontreal.ca/~andrew/PDF/icm.pdf |accessdate=2024-01-09 |archive-date=2016-05-07 |archive-url=https://web.archive.org/web/20160507093313/http://www.dms.umontreal.ca/~andrew/PDF/icm.pdf |dead-url=no }}.</ref><ref name="Pintz07">{{citation|last=Pintz|first=János|author-link=János Pintz|title=Cramér vs. Cramér: On Cramér's probabilistic model for primes|journal=Funct. Approx. Comment. Math.|volume=37|issue=2|year=2007|pages=232–471|doi=10.7169/facm/1229619660|url=http://projecteuclid.org/euclid.facm/1229619660|mr=2363833|zbl=1226.11096|s2cid=120236707|doi-access=free|accessdate=2024-01-09|archive-date=2020-06-26|archive-url=https://web.archive.org/web/20200626061241/https://projecteuclid.org/euclid.facm/1229619660|dead-url=no}}</ref>和{{link-en|赫尔穆特·迈尔 (数学家)|Helmut Maier|赫尔穆特·迈尔}}等人的直觀猜測不一致。<ref>{{link-en|Leonard Adleman|Leonard Adleman}} and Kevin McCurley, "[ftp://www.cs.sandia.gov/pub/papers/mccurley/open.ps Open Problems in Number Theoretic Complexity, II]{{Dead link}}" (PS), ''Algorithmic number theory'' (Ithaca, NY, 1994), Lecture Notes in Comput. Sci. '''877''': 291–322, Springer, Berlin, 1994.  {{doi|10.1007/3-540-58691-1_70}}. {{ISBN|978-3-540-58691-3}}.</ref><ref>{{Citation | last1=Maier | first1=Helmut | author-link=Helmut Maier | title=Primes in short intervals | url=http://projecteuclid.org/euclid.mmj/1029003189 | doi=10.1307/mmj/1029003189 | mr=783576 | year=1985 | journal=The Michigan Mathematical Journal | issn=0026-2285 | volume=32 | issue=2 | pages=221–225 | zbl=0569.10023 | doi-access=free | accessdate=2024-01-09 | archive-date=2020-08-12 | archive-url=https://web.archive.org/web/20200812011357/https://projecteuclid.org/euclid.mmj/1029003189 | dead-url=no }}</ref>而這些人的直觀猜測認為，對任意的<math>\varepsilon>0</math>下式對無限多的數成立：

:<math> g_n > \frac{2-\varepsilon}{e^\gamma}(\log p_n)^2 \approx 1.1229(\log p_n)^2 ,</math>

其中<math>\gamma</math>是[[歐拉-馬斯刻若尼常數]]。

兩個相關的猜想（可見{{OEIS2C|id=A182514}}的討論）如下：

比菲鲁兹巴赫特猜想來得弱的猜想：
:<math>\left(\frac{\log(p_{n+1})}{\log(p_n)}\right)^n < e,</math>

比菲鲁兹巴赫特猜想來得強的猜想：

:<math>\left(\frac{p_{n+1}}{p_n}\right)^n < n\log(n)\qquad \text{ for all } n > 5,</math>

==參見==
*[[質數定理]]
*[[安德里卡猜想]]
*[[勒讓德猜想]]
*[[奥珀曼猜想]]
*[[克拉梅爾猜想]]

==註解==
{{Reflist|colwidth=80em}}

==參考資料==
* {{cite book|last=Ribenboim|first=Paulo|title=The Little Book of Bigger Primes Second Edition |url=https://archive.org/details/littlebookofbigg0000ribe| publisher=Springer-Verlag | year=2004|isbn=0-387-20169-6}}
* {{cite book|last=Riesel|first=Hans|title=Prime Numbers and Computer Methods for Factorization, Second Edition | publisher=Birkhauser|year=1985|isbn=3-7643-3291-3}}

{{質數猜想}}
[[Category:素數猜想]]
[[Category:數論未解决問題]]