2

Firoozbakht’s conjecture asserts that, for every prime number: $$\sqrt[{k+1}]{p_{k+1}}\lt\sqrt[{k}]{p_k} \ \ \forall k\ge 1$$ Cramer's conjecture asserts that: $$p_{n+1}-p_n=O(\log p_n)^2$$ Firoozbakht’s conjecture is believed to be false, as it contradicts the Cramer's one. Why? Thanks in advance.

George N. Missailidis
  • 788
  • 6
  • 19
Riccardo.Alestra
  • 10,546
  • I would write out an inequality using the definition of the $O$ notation being employed in Cramer's conjecture, from there I would manipulate that inequality until I could get something similar to the first conjecture, in hopes I would encounter said contradiction. – Ethan Splaver Jun 17 '13 at 11:49
  • 3
    Gerry Myerson's answer below indicates that it's not Cramer's conjecture that it contradicts (it seems that, in fact, it implies it), but rather the Cramer-Granville heuristic. – Stephen Jun 17 '13 at 13:10
  • 1
    While I agree with the answer above, I wanted to point out a Firoozbakht’s conjecture correction. $\sqrt[{p_{k+1}}]{p_{k+1}}\lt\sqrt[{p_k}]{p_k}$ Should be $\sqrt[{k+1}]{p_{k+1}}\lt\sqrt[{k}]{p_{k}}$, or it should be $p_k^{1/k} > p_{k+1}^{1/{(k+1)}}$. – user160140 May 01 '14 at 02:09

1 Answers1

5

Copying my answer to Ratio of logarithmic primes;

This is Firoozbakht’s conjecture. According to the link, it has been verified for primes up to $4\times10^{18}$, but is believed to be false, as it contradicts the Cramér–Granville heuristic.

Added: there is also some discussion which may be useful at https://mathoverflow.net/questions/90327/any-progress-on-the-firoozbakht-conjecture

Community
  • 1
Gerry Myerson
  • 179,216
  • See also http://mathoverflow.net/questions/93643/an-empirical-observation-on-a-limit-involving-consecutive-primes/93663 – Gerry Myerson Jun 17 '13 at 13:10
  • Why do we believe that it is Firoozbakht's conjecture that should be false and not Cramer-Granville's heuristics ? – Sylvain Julien Mar 27 '17 at 18:40
  • @Sylvain Julien - Different authors believe different things; e.g. the authors of arXiv:1604.03496 believe Firoozbakht's conjecture is true. The author of arXiv:1208.2683 believes that there are at most finitely many violations of Firoozbakht's conjecture (see Conj. 2.3 in arXiv:1208.2683); that is, Firoozbakht's conjecture is true for large enough primes. – Alex Mar 27 '17 at 21:20
  • Yet another distinct possibility is that BOTH Firoozbakht's conjecture and Cramer-Granville's heuristics are false. – Alex Mar 27 '17 at 21:22
  • Another source of information on the relation between Firoozbakht and prime gaps is Alexei Kourbatov, Upper bounds for prime gaps related to Firoozbakht's conjecture, J Integer Sequences 18 (2015) 15.11.2, available at https://cs.uwaterloo.ca/journals/JIS/VOL18/Kourbatov/kourb7.pdf – Gerry Myerson Mar 27 '17 at 23:11