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 in 1982.

The conjecture states that ${\displaystyle {\sqrt[{n}]{p_{n}}}}$ (where ${\displaystyle p_{n}}$ is the ${\displaystyle n}$-th prime) is a strictly decreasing function of ${\displaystyle n}$; i.e.,

for all ${\displaystyle n\geq 1}$. Equivalently, ${\displaystyle p_{n+1}<p_{n}^{1+1/n}}$. See OEIS: A182134, OEIS: A246782.

By using a table of maximal gaps, Firoozbakht verified her conjecture up to ${\displaystyle 4.444\cdot 10^{12}}$. Now with more extensive tables of maximal gaps, the conjecture has been verified for all primes below ${\displaystyle 2^{64}\approx 1.84\cdot 10^{19}}$.

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

for all ${\displaystyle n\geqslant 5}$, and

for all ${\displaystyle n\geqslant 10}$. See also OEIS: A111943. 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

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