[seqfan] Re: Maximal prime gaps
Neil Fernandez
primeness at borve.org
Wed Feb 22 14:47:15 CET 2023
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
>> >
>> > --
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>> >
>>
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>>
>
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Neil Fernandez
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