菲鲁兹巴赫特猜想
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在數論中,菲鲁兹巴赫特猜想(Firoozbakht's conjecture 或 Firoozbakht conjecture[1][2])是數學上關於質數分布的一個猜想。該猜想以伊朗女數學家法丽德·菲鲁兹巴赫特的名字命名,她於1982年提出此猜想。
該猜想聲稱,是一個嚴格遞減函數(其中是第個質數),也就是說
或等價地
藉由使用最大質數間隙(maximal gap)表,法丽德·菲鲁兹巴赫特確認她的猜想對大到的數都成立。[2]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的質數都成立。[3][4]
此外,[6]
對此可見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。[4]從該猜想可推出強克拉梅爾猜想,而這與安德鲁·格兰维尔、平茨·亚诺什[7][8][9]和赫尔穆特·迈尔等人的直觀猜測不一致。[10][11]而這些人的直觀猜測認為,對任意的下式對無限多的數成立:
其中是歐拉-馬斯刻若尼常數。
兩個相關的猜想(可見A182514的討論)如下:
比菲鲁兹巴赫特猜想來得弱的猜想:
比菲鲁兹巴赫特猜想來得強的猜想:
參見
註解
- ^ Ribenboim, Paulo. The Little Book of Bigger Primes Second Edition
. Springer-Verlag. 2004: 185. ISBN 9780387201696.
- ^ 2.0 2.1 Rivera, Carlos. Conjecture 30. The Firoozbakht Conjecture. [22 August 2012].
- ^ Gaps between consecutive primes
- ^ 4.0 4.1 Kourbatov, Alexei. Prime Gaps: Firoozbakht Conjecture.
- ^ Sinha, Nilotpal Kanti, On a new property of primes that leads to a generalization of Cramer's conjecture, 2010, arXiv:1010.1399
[math.NT].
- ^ Kourbatov, Alexei, Upper bounds for prime gaps related to Firoozbakht's conjecture, Journal of Integer Sequences, 2015, 18 (Article 15.11.2), MR 3436186, Zbl 1390.11105, arXiv:1506.03042
.
- ^ Granville, A., Harald Cramér and the distribution of prime numbers (PDF), Scandinavian Actuarial Journal, 1995, 1: 12–28, MR 1349149, Zbl 0833.01018, doi:10.1080/03461238.1995.10413946, (原始内容 (PDF)存档于2016-05-02).
- ^ Granville, Andrew, Unexpected irregularities in the distribution of prime numbers (PDF), Proceedings of the International Congress of Mathematicians, 1995, 1: 388–399, ISBN 978-3-0348-9897-3, Zbl 0843.11043, doi:10.1007/978-3-0348-9078-6_32.
- ^ Pintz, János, Cramér vs. Cramér: On Cramér's probabilistic model for primes, Funct. Approx. Comment. Math., 2007, 37 (2): 232–471, MR 2363833, S2CID 120236707, Zbl 1226.11096, doi:10.7169/facm/1229619660
- ^ Leonard Adleman and Kevin McCurley, "Open Problems in Number Theoretic Complexity, II" (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.
- ^ Maier, Helmut, Primes in short intervals, The Michigan Mathematical Journal, 1985, 32 (2): 221–225, ISSN 0026-2285, MR 0783576, Zbl 0569.10023, doi:10.1307/mmj/1029003189
參考資料
- Ribenboim, Paulo. The Little Book of Bigger Primes Second Edition. Springer-Verlag. 2004. ISBN 0-387-20169-6.
- Riesel, Hans. Prime Numbers and Computer Methods for Factorization, Second Edition. Birkhauser. 1985. ISBN 3-7643-3291-3.