[seqfan] Re: Problem

M. F. Hasler seqfan at hasler.fr
Thu Aug 13 11:10:40 CEST 2020

On Wed, 12 Aug 2020, 10:56 Tomasz Ordowski, <tomaszordowski at gmail.com>

> 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
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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