# [seqfan] For reflection and calculations

Tomasz Ordowski tomaszordowski at gmail.com
Mon Sep 21 20:00:44 CEST 2020

```Dear readers!

By the dual Sierpinski conjecture,
there are no odd primes p such that (p-2^m)2^n+1 is composite for every
1<2^m<p and n>0.
Are there odd primes p such that (p-2^m)2^n+1 is composite for every
1<2^m<p and 1<2^n<p ?
Find odd numbers k>1 such that (k-2^m)2^n+1 is composite for every 2^m<k
and 2^n<k.
Note that such a number k (if it exists) must be composite (put m=n=0).

By the dual Riesel conjecture,
there are no odd primes p such that (p+2^m)2^n-1 is composite for every m>0
and n>0.
Are there odd primes p such that (p+2^m)2^n-1 is composite for every
1<2^m<p and 1<2^n<p ?
Find odd numbers k>1 such that (k+2^m)2^n-1 is composite for every 2^m<k
and 2^n<k.
Note that such a number k (if it exists) must be composite (put m=n=0).

Here are my conjectures in three combinations:

Conjecture 1.
(a) There are infinitely many odd numbers k>1 such that (k-2^m)2^n-1 is
composite for every 2^m<k and 2^n<k. (*)
However, (b) there are no odd numbers k>1 such that (k-2^m)2^n+1 is
composite for every 2^m<k and 2^n<k.

Conjecture 2.
(a) There are infinitely many odd numbers k>1 such that (k+2^m)2^n+1 is
composite for every 2^m<k and 2^n<k. (**)
However, (b) there are no odd numbers k>1 such that (k+2^m)2^n-1 is
composite for every 2^m<k and 2^n<k.

Conjecture 3.
(a) There are infinitely many odd naturals k that are not of the form
p+2^m+2^n. (***)
(b) There are infinitely many odd integers k that are not of the form
p-2^m-2^n.
However, (c) every odd integer k is of the form p+2^m-2^n or -p-2^m+2^n.
Where p is a prime.
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Note: for m>=0 and n>=0 in all three of the above conjectures.

These conjectures are an attempt to solve the problems
and answer the questions from my previous post.

Best regards,

Thomas Ordowski
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(*) There are odd numbers k>1 such that
(k-2^m)2^n-1 is composite for every 2^m<k and 2^n<k,
namely 65535 and 13975275 [found by Amiram Eldar].
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(**) There are odd numbers k>1 such that
(k+2^m)2^n+1 is composite for every 2^m<k and 2^n<k,
namely 14568915 [Amiram Eldar found only this one].
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