Firoozbakht's conjecture

In number theory, Firoozbakht's conjecture (or the Firoozbakht conjecture ) 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 $$p_{n}^{1/n}$$ (where $$p_n$$ is the nth prime) is a strictly decreasing function of n, i.e.,


 * $$\sqrt[n+1]{p_{n+1}} < \sqrt[n]{p_n} \qquad \text{ for all } n \ge 1.$$

Equivalently:


 * $$p_{n+1} < p_n^{1+\frac{1}{n}} \qquad \text{ for all } n \ge 1,$$

see,.

By using a table of maximal gaps, Farideh Firoozbakht verified her conjecture up to 4.444. Now with more extensive tables of maximal gaps, the conjecture has been verified for all primes below 264 &asymp; $1.84$.

If the conjecture were true, then the prime gap function $$g_n = p_{n+1} - p_n $$ would satisfy:


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

Moreover:


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

see also. This is among the strongest upper bounds conjectured for prime gaps, even somewhat stronger than the Cramér and Shanks conjectures. It implies a strong form of Cramér's conjecture and is hence inconsistent with the heuristics of Granville and Pintz and of Maier  which suggest that


 * $$ g_n > \frac{2-\varepsilon}{e^\gamma}(\log p_n)^2 \approx 1.1229(\log p_n)^2 ,$$

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

Two related conjectures (see the comments of ) are


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

which is weaker, and


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

which is stronger.