OFFSET
1,1
COMMENTS
The Firoozbakht conjecture: (prime(n+1))^(1/(n+1)) < prime(n)^(1/n), or prime(n+1) < prime(n)^(1+1/n), prime(n+1)/prime(n) < prime(n)^(1/n), (prime(n+1)/prime(n))^n < prime(n).
Using the Mathematica program shown below, I have found no further terms below 2^27. I conjecture that this sequence is finite and that the terms stated are the only members. - Robert G. Wilson v, May 06 2012 [Warning: this conjecture may be false! - N. J. A. Sloane, Apr 25 2014]
I conjecture the contrary: the sequence is infinite. Note that 10^13 < a(6) <= 1693182318746371. - Charles R Greathouse IV, May 14 2012
[Stronger than Firoozbakht] conjecture: All (prime(n+1)/prime(n))^n values, with n >= 5, are less than n*log(n). - John W. Nicholson, Dec 02 2013, Oct 19 2016
The Firoozbakht conjecture can be rewritten as (log(prime(n+1)) / log(prime(n)))^n < (1+1/n)^n. This suggests the [weaker than Firoozbakht] conjecture: (log(prime(n+1))/log(prime(n)))^n < e. - Daniel Forgues, Apr 26 2014
The inequality in the definition is equivalent to the inequality prime(n+1)-prime(n) > log(n)*log(prime(n)) for sufficiently large n. - Thomas Ordowski, Mar 16 2015
Prime indices, A000720(a(n)) = 1, 2, 4, 30, 217, 49749629143526. - John W. Nicholson, Oct 25 2016
REFERENCES
Farhadian, R. (2017). On a New Inequality Related to Consecutive Primes. OECONOMICA, vol 13, pp. 236-242.
LINKS
Reza Farhadian, A New Conjecture On the primes, Preprint, 2016.
R. Farhadian, and R. Jakimczuk, On a New Conjecture of Prime Numbers Int. Math. Forum, vol. 12, 2017, pp. 559-564.
Luan Alberto Ferreira, Some consequences of the Firoozbakht's conjecture, arXiv:1604.03496v2 [math.NT], 2016.
Luan Alberto Ferreira, Hugo Luiz Mariano, Prime gaps and the Firoozbakht Conjecture, São Paulo J. Math. Sci. (2018), 1-11.
A. Kourbatov, Verification of the Firoozbakht conjecture for primes up to four quintillion, arXiv:1503.01744 [math.NT], 2015.
A. Kourbatov, Upper bounds for prime gaps related to Firoozbakht's conjecture, J. Int. Seq. 18 (2015) 15.11.2.
Carlos Rivera, Conjecture 78. P_n^((P_n+1/P_n)^n) <= n^P_n, 2016.
Nilotpal Kanti Sinha, On a new property of primes that leads to a generalization of Cramer's conjecture, arXiv:1010.1399 [math.NT], 2010.
Matt Visser, Verifying the Firoozbakht, Nicholson, and Farhadian conjectures up to the 81st maximal prime gap, arXiv:1904.00499 [math.NT], 2019.
Wikipedia, Firoozbakht’s conjecture
EXAMPLE
7 is in the list because, being the 4th prime, and 11 the fifth prime, we verify that (11/7)^4 = 6.09787588507... which is greater than 4.
11 is not on the list because (13/11)^5 = 2.30543740804... and that is less than 5.
MATHEMATICA
Prime[Select[Range[1000], (Prime[# + 1]/Prime[#])^# > # &]] (* Alonso del Arte, May 04 2012 *)
firoozQ[n_, p_, q_] := n * Log[q] > Log[n] + n * Log[p]; k = 1; p = 2; q = 3; While[ k < 2^27, If[ firoozQ[k, p, q], Print[{k, p}]]; k++; p = q; q = NextPrime@ q] (* Robert G. Wilson v, May 06 2012 *)
PROG
(PARI) n=1; p=2; forprime(q=3, 1e6, if((q/p*1.)^n++>n, print1(p", ")); p=q) \\ Charles R Greathouse IV, May 14 2012
(PARI) for(n=1, 75, if((A000101[n]/A002386[n]*1.)^A005669[n]>=A005669[n], print1(A002386[n], ", "))) \\ Each sequence is read in as a vector as to overcome PARI's primelimit \\ John W. Nicholson, Dec 01 2013
(PARI) q=3; n=2; forprime(p=5, 10^9, result=(p/q)^n/(n*log(n)); if(result>1, print(q, " ", p, " ", n, " ", result)); n++; q=p) \\ for stronger than Firoozbakht conjecture \\ John W. Nicholson, Mar 16 2015, Oct 19 2016
CROSSREFS
KEYWORD
nonn
AUTHOR
Thomas Ordowski, May 04 2012
EXTENSIONS
a(6) from John W. Nicholson, Dec 01 2013
STATUS
approved