[seqfan] Re: Maximal prime gaps

[Neil Fernandez]

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/
>>
>
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-- 
Neil Fernandez