Firoozbakht's conjecture

Prime gap function

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 from the University of Isfahan who stated it first in 1982.

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

$p_{n+1}^{1/(n+1)} < p_n^{1/n} \text{ for all } n \ge 1.$

Equivalently: $p_{n+1} < p_n^{1+\frac{1}{n}}\text{ for all } n \ge 1,$ see  A182134,  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 4×10¹⁸.^[3]

If the conjecture is true, then the prime gap function $g_n = p_{n+1} - p_n$ satisfies $g_n < (\log p_n)^2 - \log p_n \text{ for all } n > 4$^[4] and, moreover, $g_n < (\log p_n)^2 - \log p_n - 1 \text{ for all } n > 9$;^[5] see also  A111943. This is one of the strongest upper bound ever conjectured for prime gaps, even somewhat stronger than the Cramer and Shanks conjectures.^[6] It implies a strong form of Cramér's conjecture but is inconsistent with the heuristics of Granville and Pintz^[7][8][9] which suggest that $g_n > \frac{2-\varepsilon}{e^\gamma}(\log p_n)^2$ infinitely often for any $\varepsilon>0,$ where $\gamma$ denotes the Euler–Mascheroni constant.

Two related conjectures (see the comments of  A182514) are

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

which is weaker, and

$\left(\frac{p_{n+1}}{p_n}\right)^n < n\log(n)$ for all values with $n > 5$,

which is stronger.

See also

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

Notes

1. Ribenboim, Paulo. The Little Book of Bigger Primes Second Edition. Springer-Verlag. p. 185.
2. Rivera, Carlos. "Conjecture 30. The Firoozbakht Conjecture". Retrieved 22 August 2012.
3. Gaps between consecutive primes
4. Sinha, Nilotpal Kanti (2010), On a new property of primes that leads to a generalization of Cramer's conjecture, pp. 1–10, arXiv:1010.1399.
5. Kourbatov, Alexei (2015), "Upper bounds for prime gaps related to Firoozbakht’s conjecture", Journal of Integer Sequences 18 (Article 15.11.2), arXiv:1506.03042.
6. Kourbatov, Alexei. "prime Gaps: Firoozbakht Conjecture".
7. Granville, A. (1995), "Harald Cramér and the distribution of prime numbers" (PDF), Scandinavian Actuarial Journal 1: 12–28.
8. Granville, Andrew (1995), "Unexpected irregularities in the distribution of prime numbers" (PDF), Proceedings of the International Congress of Mathematicians 1: 388–399.
9. Pintz, János (2007), "Cramér vs. Cramér: On Cramér's probabilistic model for primes", Funct. Approx. Comment. Math. 37 (2): 232–471

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.