Firoozbakht's conjecture

Prime gap function

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 $\sqrt[n]{p_n}$ (where $p_n$ is the $n$-th prime) is a strictly decreasing function of $n$; i.e.,

$\sqrt[n+1]{p_{n+1}}<\sqrt[n]{p_n}$

for all $n\ge 1$. Equivalently, $p_{n+1}<p_n^{1+1/n}$. See Template:OEIS2C, Template:OEIS2C.

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

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

$g_n<(\log p_n{)}^2-\log p_n$

for all $n⩾5$, and

$g_n<(\log p_n{)}^2-\log p_n-1$

for all $n⩾10$. See also Template:OEIS2C. 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

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

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

Three related conjectures (see the comments of Template:OEIS2C) are variants of Firoozbakht's. Forgues notes that Firoozbakht's can be written

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

where the right hand side is the well-known expression which reaches Euler's number in the limit $n\to \mathrm{\infty }$, suggesting the slightly weaker conjecture

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

Nicholson and Farhadian give two stronger versions of Firoozbakht's conjecture which can be summarized as:

${\left(\frac{p_{n+1}}{p_n}\right)}^n<\frac{p_n\log n}{\log p_n}<n\log n<p_n\text{ for all }n⩾6,$

where the right-hand inequality is Firoozbakht's, the middle is Nicholson's (since $n\log n<p_n$; see the article on non-asymptotic bounds on the prime-counting function) and the left-hand inequality is Farhadian's (since $p_n/\log p_n<n$; see prime-counting function § inequalities.

All have been verified to 2⁶⁴.

See also

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

References

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