Problems & Puzzles: Conjectures

Conjecture 78. Pn^((Pn+1/Pn)^n)<=n^Pn

Reza Farhadian, from Lorestan Univesity, in Lorestan Iran, sent the following conjecture:

Pn^((Pn+1/Pn)^n)<=n^Pn

Reza has confirmed his own conjecture "for the first 10^4 primes".

Reza makes the following claim:

His conjecture is stronger than the Nicholson's conjecture (See the Nicholson's conjecture here or there)

It has already been proved that several conjectures about the same issue are relatively stronger, according this scheme:

Nicholson > Firoozbakht > Cramer >  Granville

Accordingly, The Reza's conjecture is stronger than all of these.

In order that you may read Reza's claim in his own words, please read directly his paper here.

Q1. Can you confirm the Reza's conjecture for a larger quantity of primes?

Q2. Independently of the strongest character of the Reza's conjecture -related to the other four conjectures mentioned- its mathematical form, makes easier or harder to prove it, if this might be possible in the future?

Contribution came from Emmanuel Vantieghem

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Emmanuel wrote:

Q1.
I tested the double log form of the conjecture, i.e. :
n ( log q - log p ) - log ( p log n) - log(log(n)) <= 0,   (*)
where p is the n_th prime and  q the next prime to p.
I did that for every  p  such that  q - p  is a record gap.
I took  p  in the list of maximal gaps that appeared in  Wikipedia, Prime gap.
The left hand member (LHM) of (*) was negative for all  p.
For values of  p  for which the computation of  pi(p)  was impossible with Mathematica, I used the inequality of Dusart :
x ( log x + 1)/(log x)^2 < pi(x) < x((log x)^2 + log x + 2.51)/(log x)^3
In those cases I got two negative values of the LHM between which the true value of the LHM  must be found.
So, I think the conjecture is true for all  p <= 1425172824437699411 (and maybe for all  p <= the still to recover next prime in that table).

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John W. Nicholson sent the following link:

A Conjecture Sharper than Cramér's and Firoozbakht's

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On April 3, 2017, Reza Farhadian wrote:

Carlos, within the conjecture 78, please put the strictly form of conjectural inequality, because the equality state (=) never is not occur for finite integer n. So, put the following form of inequality
(p_n)^((p_(n+1)/p_n )^n )<n^(p_n ).

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On April 15, 2017 Reza wrote again:

Dr. L. A. Ferreira and H. L. Mariano, proved that

Farhadian’s conjecture =>Nicholson’s conjecture =>Firoozbakht’s conjecture => Forgues’s conjecture

For their main work, see last version of Ferreira’s article available at http://arxiv.org/abs/1604.03496v2.

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Charles Greathouse wrote:

The 76th and 77th prime gaps have now been discovered and I verified that Farhadian's conjecture holds for them, and hence for all primes from the 5th to the 10^17th (in fact the 77th prime gap starts at prime(160332893561542066) = 6787988999657777797).

Of course modern heuristics suggest that the conjectures of Farhadian, Nicholson, Firoozbakht, Forgues, and the strong form of the Cramér conjecture all fail infinitely often, but it's not likely that a counterexample will be found as they should be quite sparse.

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