# Firoozbakht’s conjecture

Prime gap function

In number theory, Firoozbakht’s conjecture, also known as the Firoozbakht conjecture,[1][2][3] states that $p_{n}^{1/n}\,$ (where $p_n\,$ is the nth prime) is a strictly decreasing function of n, i.e.,

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

Equivalently: $p_{n+1} < p_{n}^{1+1/n}$ for all $n \ge 1.$

Another equivalent statement, $\left(\frac{p_{n+1}}{p_{n}}\right)^n < p_{n}$ for all $n \ge 1,$ is the motivating reason for the sequence .

The conjecture is named after Farideh Firoozbakht, from the University of Isfahan, who stated it in 1982. If this conjecture is true, then the prime gap function $g_n = p_{(n+1)} - p_{n}$ satisfies $g_n < (\log p_{n})^2 - \log p_{n}$ which is sharper than Cramér's conjecture $g_n = O((\log p_n)^2).$ Currently, the largest stated verification was done by using "Maximal gaps between consecutive primes less than 4.444 * 1012" by Firoozbakht. Using a table of maximal gaps and the above gap inequality, the confirmation value can be extended to all primes below 4×1018.[4]

The conjecture is believed to be false, as it contradicts the Cramér–Granville heuristic. If it were true it would imply the weaker Cramér conjecture.