## October 30, 2021

### Firoozbakht’s Conjecture

#### Posted by John Baez

The Iranian mathematician Farideh Firoozbakht made a strong conjecture in 1982: the $n$th root of the $n$th prime keeps getting smaller as we make $n$ bigger! For example:

$\sqrt[1]{2} \; > \; \sqrt[2]{3} \; > \; \sqrt[3]{5} \; > \; \sqrt[4]{7} \; > \; \sqrt[5]{11} \; > \; \cdots$

It’s been checked for all primes up to about 18 quintillion, but nobody has a clue how to prove it. In fact, some experts think it’s probably false.

Firoozbakht’s conjecture is a way of saying the gaps between successive primes don’t get too big too fast. As you can see here, it’s stronger than Cramér’s or Granville’s conjectures on prime gaps — and it gets scarily close to being wrong at times, like when the largest prime gap jumps from 924 to 1132 at the prime 1693182318746371 — you can see that big horizontal black line above.

Farideh Firoozbakht checked her conjecture up to about 4 trillion using a table of large gaps between primes: those are what could make the conjecture false. In 2015 Alexei Kourbatov checked it up to 4 quintillion, in this paper here:

Now Kourbatov claims to have checked Firoozbakht’s conjecture for all primes less than $2^64$, which is a bit over 18 quadrillion. The work is on a website that links to someone’s table of big prime gaps, and the link seems a bit broken:

Maybe this table of prime gaps will do the job.

If true, Firoozbakht’s conjecture will imply a lot of good stuff, listed in this paper by Ferreira and Mariano:

For example: there are at least 2 primes between consecutive square numbers, and at least 4 between consecutive cubes.

It’s known that the Riemann Hypothesis implies the $n$th prime gap is eventually less than $\sqrt{p_n} \, \ln p_n$, where $p_n$ is the $n$th prime. Firoozbakht’s conjecture implies something much stronger: it’s eventually less than

$c(\ln p_n)^2$

where you can choose $c$ to be any constant bigger than 1. This is Theorem 2.2 in Ferreira and Mariano’s paper.

Nobody knows how to prove Firoozbakht’s conjecture using the Riemann Hypothesis. But conversely, it seems nobody has shown Firoozbakht implies Riemann!

There are even some good theoretical reasons for believing Firoozbakht’s conjecture is false! Granville has a nice probabilistic model of the behavior of primes that predicts that there are infinitely many gaps arbitrarily close to

$2e^{-\gamma} \, (\ln p_n)^2 \approx 1.1229 \, (\ln p_n)^2$

in width. Firoozbakht’s conjecture, on the other hand, implies they’re eventually all significantly smaller than this, as I mentioned above.

For more on Granville’s probabilistic model of primes and its consequences for the largest prime gaps, see:

Farideh Firoozbakht, alas, died in 2019 at the age of 57. She studied pharmacology and later mathematics at the University of Isfahan, and later taught mathematics at that university. That, and her conjecture, is all I know about her. I wish I could ask her how she came up with her conjecture, and ask her a bit about what her life was like.

If anyone knows, please tell me!

I thank Maarten Morier for telling me about Firoozbakht’s conjecture.

Posted at October 30, 2021 2:35 AM UTC

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### Re: Firoozbakht’s Conjecture

There is a little more about Farideh Firoozbakht at Carlos Rivera’s site Prime Puzzles and Problem Connection.

Posted by: Richard Pinch on October 30, 2021 9:16 AM | Permalink | Reply to this

### Re: Firoozbakht’s Conjecture

Interesting, thanks!

I was born in 1962 in Isfahan, Iran. Since I was seventeen, because of my deep affection for the animals, I became a vegetarian.

After graduating from high school, I went to the University of Isfahan to continue my studies in pharmacology which is one of the most favored disciplines in Iran. But, since I was very fond of mathematics, especially number theory, I changed my major to mathematics in the third year and I became graduated in 1987.

Afterwards, I went to Isfahan University of Technology to continue my postgraduate studies in mathematics.

Since 1992, I have been instructing mathematics at the Iranian universities and currently I am teaching at the University of Isfahan.

I have done research in number theory and I have been able to find beautiful and interesting relations. My favorite field is “The Gap between Prime Numbers.”

Conjecture No. 30 came to mind in 1982, while I was studying the proof of the “Prime Number Theorem”. I think that the conjecture is the most important conjecture related to prime numbers, it shows one of the most interesting and beautiful behaviors of the prime numbers and I believe working on this conjecture could guide us to important results in number theory.

Posted by: John Baez on October 30, 2021 7:08 PM | Permalink | Reply to this

### Re: Firoozbakht’s Conjecture

Any idea how much is known rigorously about the sequence $p_n^{1/n}$?

Posted by: Mark Meckes on October 30, 2021 1:14 PM | Permalink | Reply to this

### Re: Firoozbakht’s Conjecture

A classic paper with explicit bounds for $p_n$ and many other prime-related functions is “Approximate formulas for some functions of prime numbers”, J. Barkley Rosser, Lowell Schoenfeld, Illinois J. Math. 6(1): 64-94 (March 1962). DOI: 10.1215/ijm/1255631807

One example: $n(\log n + \log\log n - \frac{3}{2}) \lt p_n \lt n(\log n + \log\log n - \frac{1}{2})$ for $n \ge 20$

Posted by: Richard Pinch on October 30, 2021 5:28 PM | Permalink | Reply to this

### Re: Firoozbakht’s Conjecture

Thanks. So that would give some pretty tight estimates on how quickly $p_n^{1/n}$ approaches 1, for instance. And it’s clearly always greater than 1, so it’s typically decreasing in some sense. But both of those facts would be consistent with, say, $p_n^{1/n}$ increasing at every third $n$ for $p_n \ge 2^{100}$.

Is there anything proven more directly in the direction of this conjecture, say limiting the density of those $n$ where $p_n^{1/n}$ increases?

Posted by: Mark Meckes on October 30, 2021 6:19 PM | Permalink | Reply to this

### Re: Firoozbakht’s Conjecture

In my studies so far I’ve seen very little about the tendency for $p_n^{1/n}$ to decrease. People instead seem mainly to prove bounds on $p_n$ (as above) and on the prime gaps $g_n = p_{n+1} - p_n$.

The only interesting work on the tendency for $p_n^{1/n}$ to decrease that I know is the technique Alexei Kourbatov used to efficiently prove that $p_n^{1/n}$ is strictly decreasing for $p_n \le 4 \cdot 10^{18}$:

This paper is quite elementary and easy to read.

Posted by: John Baez on October 30, 2021 8:19 PM | Permalink | Reply to this

### Re: Firoozbakht’s Conjecture

I do not claim that Firoozbakht’s conjecture follows almost surely from Cramer’s model; far from that. My claim is much weaker: in Cramer’s model, almost all maximal gaps obey Firoozbakht’s inequality. Note that the sequence of maximal gaps is infinite (with probability 1). So I actually only claim that in Cramer’s model, the “Firoozbakht violator gaps” form a thin subsequence among all maximal gaps, also with probability 1. The actual wording in my paper is “all maximal gap sizes are below log n(log n - 1), except for a vanishing proportion of maximal gaps.” This vanishing proportion is expected to constitute still an infinite (but very thin) subsequence among maximal gaps in Cramer’s model.

Posted by: Alexei Kourbatov on October 30, 2021 8:18 PM | Permalink | Reply to this

### Re: Firoozbakht’s Conjecture

Thanks! You seem to be correcting a misunderstanding I expressed on Twitter, not here. I’m glad to have it cleared up.

Posted by: John Baez on October 30, 2021 8:28 PM | Permalink | Reply to this

### Re: Firoozbakht’s Conjecture

Regarding the tendency of pk1/k to decrease - here is a related theorem.

Theorem (arXiv:1506.03042; J. Integer Seq., 2015, 15.11.2, Thm.5):

Let pk be the k-th prime, then for large k we have

pk1+1/kpk = ln2 pk − ln pk − 1 + o(1).

Posted by: Alexei Kourbatov on October 30, 2021 9:21 PM | Permalink | Reply to this

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