# [seqfan] Re: Maximal prime gaps

Tomasz Ordowski tomaszordowski at gmail.com
Wed Feb 22 16:46:10 CET 2023

```Hello Neil.

Thank you for reminding me of my sequence,
where Forgues' conjecture is in the equivalent form.

Thomas

śr., 22 lut 2023 o 14:47 Neil Fernandez <primeness at borve.org> napisał(a):

> Hi Tomasz and everyone.
>
> That was conjectured by Daniel Forgues in 2014: see comment on A182514.
>
> Among the first million primes, the only cases for which
>
> k := n* ( log log prime(n+1) - log log prime(n) ) > 0.7
>
> are
>
> n               prime(n)                k
>
> 2               3                       0.763674
> 4               7                       0.835446
> 30              113                     0.7322
> 217             1327                    0.762132
> 3385            31397                   0.748738
> 31545           370261                  0.744072
> 40933           492113                  0.723392
> 104071          1357201                 0.716759
> 118505          1561919                 0.702214
> 149689          2010733                 0.759089
> 325852          4652353                 0.70254
>
> There is also
>
> 49749629143526          1693182318746371        0.950011
>
> (Wow!)
>
>
> Neil
>
>
> In message <CAF0qcNM_XeEuXfKYC5A+yqnTCkSp9CPJTV+KXZi_3dKnp9w42w at mail.gma
> il.com>, Tomasz Ordowski <tomaszordowski at gmail.com> writes
> >Yes ...
> >
> >Similar conjecture:
> >
> >log log prime(n+1) - log log prime(n) < 1/n.
> >
> >T. Ordowski
> >
> >wt., 21 lut 2023 o 04:18 Trizen <trizenx at gmail.com> napisa (a):
> >
> >> Just an observation:
> >>
> >> Limit_{p -> Infinity} (p^(p^(1/p)) - p - log(p)^2) = 0
> >>
> >> Therefore, if a counter-example exists, it should be just as hard to
> find
> >> as finding a counter-example to Cramer's conjecture.
> >>
> >> On Mon, Feb 20, 2023 at 11:36 AM Tomasz Ordowski <
> tomaszordowski at gmail.com
> >> >
> >> wrote:
> >>
> >> > Hello!
> >> >
> >> > Conjecture:
> >> > if p > 3, then q < p^(p^(1/p)),
> >> > where q is the next prime after p.
> >> >
> >> >   Is there a counterexample?
> >> >
> >> > Similar to Firoozbakht's conjecture,
> >> > this is close to Cramer's conjecture:
> >> > q - p = O(log p)^2.
> >> >
> >> > Best,
> >> >
> >> > Thomas
> >> > ______________________
> >> > https://en.wikipedia.org/wiki/Prime_gap
> >> >
> >> > --
> >> > Seqfan Mailing list - http://list.seqfan.eu/
> >> >
> >>
> >> --
> >> Seqfan Mailing list - http://list.seqfan.eu/
> >>
> >
> >--
> >Seqfan Mailing list - http://list.seqfan.eu/
>
> --
> Neil Fernandez
>
> --
> Seqfan Mailing list - http://list.seqfan.eu/
>
```