**Conjecture 78. ****P**_{n}^((P_{n+1}/P_{n})^n)<=n^P_{n}

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

**P**_{n}^((P_{n+1}/P_{n})^n)<=n^P_{n}

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

***

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.

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).

***

John W. Nicholson sent the following
link:

A Conjecture Sharper than Cramér's and Firoozbakht's

***

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 ).

***

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.

***

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.

***