# [seqfan] by Sierpinski and Riesel dual conjectures

Tomasz Ordowski tomaszordowski at gmail.com
Sat Jul 23 16:57:00 CEST 2022

```Dear readers,

I have something new ...

The Conditional Theorem.
By the dual Sierpinski conjecture
and by the dual Riesel conjecture;
if p is an odd prime and m is a positive integer,
then there exists n such that |(p-/+2^m)2^n+/-1| is prime.
Hard question: are the dual conjectures provable?
These conjectures condition the above theorem.

The Open Problem.
Are there odd (composite) numbers k such that
both |(k-/+2^m)2^n+/-1| are composite
for every pair of positive integers m,n ?
By the dual conjectures,
these are odd numbers k such that
both ||k-/+2^m|+/-2^n| are composite for m,n > 0.

Let's define an auxiliary sequence ...
Let a(n) be the smallest odd k such that k+2^m
is a de Polignac number P from m = 1 to n;
i.e., P-2^i is not prime for every 0 < 2^i < P.
DATA: 125, 903, 7385, 87453, 957453, 6777393,
21487809, 27035379, 1379985537, 5458529139,
15399643917, 32702289081, ...  A355885 (my draft).
Data from Amiram Eldar. I am asking for more terms.
Working conjecture: this sequence is infinite
and is bounded, namely a(n) = K for all n >= N.
Note that, for such K, each positive value of
K+2^m-2^n is composite for every m > 0.
The number K can be a (partial) solution
to the Open Problem [sic].

Best,

T. Ordowski
_____________________
Cf. A156695 and A337487.
See all three of my new drafts:
The On-Line Encyclopedia of Integer Sequences® (OEIS®)
<https://oeis.org/draft?user=Thomas%20Ordowski>

```