[seqfan] Re: Problem

[Tomasz Ordowski]

P.S. Correction.

My conditional proof is wrong!
There are odd primes q such that (q-2^n)2^n+1 is composite for every 1 <
2^n < q,
namely q = 229, 509, 523, 1181, 1481, 1489, 1657, 1709, 1867, 2179, 2371,
3061, ...
Data from Amiram Eldar.

T. Ordowski

czw., 13 sie 2020 o 11:11 M. F. Hasler <seqfan at hasler.fr> napisał(a):

> On Wed, 12 Aug 2020, 10:56 Tomasz Ordowski, <tomaszordowski at gmail.com>
> wrote:
>
> > Dear readers!
> > Are there infinitely many primes p = k2^n+1 with k odd such that q =
> k+2^n
> > is prime?
> >
>
> This looks very probable.
> It seems that about 1 / log_10(N) of all odd numbers below 2N have this
> property:
> e.g., there are 7, 51, 364, 2675, 20668, 167185,... such odd numbers 2m+1
> with m < 10, 100, 10^3, 10^4, 10^5, 10^6, ...
> And these are indeed mostly the primes, but there are also composite
> numbers and not all primes are there.
>
> Can you prove it unconditionally?
> >
>
> I didn't try this so far. I suspect that it might be similar to the twin
> prime or prime tuple conjectures where evidence is overwhelming but no
> proof is known so far.
>
>
> > Most primes seem to have this property, but it is the illusory "law of
> > small numbers", I think.
> >
>
> I don't think so. Certainly their density decreases, but even if this is
> far from infinity, I checked that up to 10^99 the next larger prime with
> this property is always not very far.
> E.g., 10^99 + 2191 has the property and only 4 smaller primes >1e99  don't
> have it..
>
> I propose some related drafts, maybe some of them get accepted:
>
> oeis.org/draft/A332075 : odd numbers 2n+1 such that k + 2^m is prime,
> where
> k and m are the odd part and 2-valuation, respectively, of 2n.
> oeis.org/draft/A332076 : corresponding indices n
> oeis.org/draft/A332078 : primes which are not in A332075
>
> Maybe I will propose another draft related to the number of primes > 2^n or
> 10^n that have to be skipped before finding a prime with that property.
>
> - Maximilian
>
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