Firoozbakht's conjecture

In number theory, Firoozbakht's conjecture (or the Firoozbakht conjecture^[1][2]) is a conjecture about the distribution of prime numbers. It is named after the Iranian mathematician Farideh Firoozbakht who stated it first in 1982.

The conjecture states that ${\displaystyle p_{n}^{1/n}}$ (where ${\displaystyle p_{n}}$ is the nth prime) is a strictly decreasing function of n, i.e.,

${\displaystyle {\sqrt[{n+1}]{p_{n+1}}}<{\sqrt[{n}]{p_{n}}}\qquad {\text{ for all }}n\geq 1.}$

Equivalently:

${\displaystyle p_{n+1}<p_{n}^{1+{\frac {1}{n}}}\qquad {\text{ for all }}n\geq 1,}$

see OEIS: A182134, OEIS: A246782.

By using a table of maximal gaps, Farideh Firoozbakht verified her conjecture up to 4.444×10¹².^[2] Now with more extensive tables of maximal gaps, the conjecture has been verified for all primes below 2⁶⁴ ≈ 1.84×10¹⁹.^[3][4]

If the conjecture were true, then the prime gap function ${\displaystyle g_{n}=p_{n+1}-p_{n}}$ would satisfy:^[5]

${\displaystyle g_{n}<(\log p_{n})^{2}-\log p_{n}\qquad {\text{ for all }}n>4.}$

Moreover:^[6]

${\displaystyle g_{n}<(\log p_{n})^{2}-\log p_{n}-1\qquad {\text{ for all }}n>9,}$

see also OEIS: A111943. This is among the strongest upper bounds conjectured for prime gaps, even somewhat stronger than the Cramér and Shanks conjectures.^[4] It implies a strong form of Cramér's conjecture and is hence inconsistent with the heuristics of Granville and Pintz^[7][8][9] and of Maier^[10][11] which suggest that

${\displaystyle g_{n}>{\frac {2-\varepsilon }{e^{\gamma }}}(\log p_{n})^{2}\approx 1.1229(\log p_{n})^{2},}$

occurs infinitely often for any ${\displaystyle \varepsilon >0,}$ where ${\displaystyle \gamma }$ denotes the Euler–Mascheroni constant.

Two related conjectures (see the comments of OEIS: A182514) are

${\displaystyle \left({\frac {\log(p_{n+1})}{\log(p_{n})}}\right)^{n}<e,}$

which is weaker, and

${\displaystyle \left({\frac {p_{n+1}}{p_{n}}}\right)^{n}<n\log(n)\qquad {\text{ for all }}n>5,}$

which is stronger.

See also

• Prime number theorem
• Andrica's conjecture
• Legendre's conjecture
• Oppermann's conjecture
• Cramér's conjecture

Notes

References

• Ribenboim, Paulo (2004). The Little Book of Bigger Primes Second Edition. Springer-Verlag. ISBN 0-387-20169-6.
• Riesel, Hans (1985). Prime Numbers and Computer Methods for Factorization, Second Edition. Birkhauser. ISBN 3-7643-3291-3.

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Mathematics in Iran

Mathematicians

Before 20th Century

Abu al-Wafa' Buzjani Jamshid al-Kashi (al-Kashi's theorem) Omar Khayyam (Khayyam-Pascal's triangle, Khayyam-Saccheri quadrilateral, Khayyam's Solution of Cubic Equations) Al-Mahani Muhammad Baqir Yazdi Nizam al-Din al-Nisapuri Al-Nayrizi Kushyar Gilani Ayn al-Quzat Hamadani Al-Isfahani Al-Isfizari Al-Khwarizmi (Al-jabr) Najm al-Din al-Qazwini al-Katibi Nasir al-Din al-Tusi Al-Biruni

Modern

Maryam Mirzakhani Caucher Birkar Sara Zahedi Farideh Firoozbakht (Firoozbakht's conjecture) S. L. Hakimi (Havel–Hakimi algorithm) Siamak Yassemi Freydoon Shahidi (Langlands–Shahidi method) Hamid Naderi Yeganeh Esmail Babolian Ramin Takloo-Bighash Lotfi A. Zadeh (Fuzzy mathematics, Fuzzy set, Fuzzy logic) Ebadollah S. Mahmoodian Reza Sarhangi (The Bridges Organization) Siavash Shahshahani Gholamhossein Mosaheb Amin Shokrollahi Reza Sadeghi Mohammad Mehdi Zahedi Mohsen Hashtroodi Hossein Zakeri Amir Ali Ahmadi

Prize Recipients

Fields Medal	Maryam Mirzakhani (2014) Caucher Birkar (2018)
EMS Prize	Sara Zahedi (2016)
Satter Prize	Maryam Mirzakhani (2013)

Organizations

• Iranian Mathematical Society

Institutions

• Institute for Research in Fundamental Sciences