[seqfan] Solving the problem with de Polignac numbers

[Tomasz Ordowski]

Dear readers,

I found (in the OEIS) many solutions to the original problem:
Are there de Polignac numbers k>1 such that k-2^n is a de Polignac number
for every 1<2^n<k ?
These are odd integers k>3 that are not of the form p+2^m+2^n with m,n>=0,
where p is a prime.
See my draft:
http://oeis.org/draft/A337487
http://oeis.org/history/view?seq=A337487&v=16

The next problem:
Are there de Polignac numbers k such that k+2^n is a composite de Polignac
number for every n>0 ?
These are odd integers k>1 that are not of the form p+2^m-2^n, where p is a
prime.
Odd integers k>1 such that every positive value of k-2^m+2^n is not prime.
If such a number k exists, it is a Sierpinski number.

Best regards,

Thomas Ordowski