Two Dual Problems

[Tomasz Ordowski]

Hello again (despite premature resignation due to AI)! 

The most mysterious (for me) questions in number theory 

are the dual problems of Sierpiński and Riesel.

The dual Sierpiński problem: is every Sierpiński number S 

(having a covering set) dual (in the following sense)? 

An odd number S is dual  

if and only if for every integer n > 0,  

both S 2^n + 1 and S + 2^n are composite.  

By the dual Sierpiński conjecture (yes), 

if p is an odd prime, then for every 1 < 2^n < p, 

there exists m > 0 such that (p - 2^n)2^m + 1 is prime, 

because p - 2^m is not a Sierpiński number (by definition). 

The dual Riesel problem: is every Riesel number R 

(having a covering set) dual (in the following sense)? 

An odd number R is dual  

if and only if for every integer n > 0, 

both R 2^n - 1 and |R - 2^n| are composite.

By the dual Riesel conjecture (yes), 

if p is an odd prime, then for every n > 0, 

there exists m > 0 such that (p + 2^n)2^m - 1 is prime, 

because (p + 2^n) is not a Riesel number (by definition). 

Note that the same applies to (2^n - p)2^m - 1 for 2^n > p.

What is the theoretical (heuristic) status 

and research progress of these problems? 

If both conjectures are true, 

then we have the (two-in-one) formula

for primes of the form q = |(p -/+ 2^n)2^m +/- 1|.  

What do you think about this?

With best regards, 

Tom Ordo (without AI assistance). 

PS. On the way to (more serious) heuristics, let us note that 

k 2^-n +- 1 = (k +- 2^n)/2^n and k +- 2^-n = (k 2^n +- 1)/2^n. 

____________

https://oeis.org/A076336/a076336c.html

https://en.wikipedia.org/wiki/Riesel_number#The_dual_Riesel_problem

[Gareth McCaughan]

On 30/01/2026 11:42, Tomasz Ordowski wrote:
> Hello again (despite premature resignation due to AI)!
>
> The most mysterious (for me) questions in number theory
> are the dual problems of Sierpiński and Riesel.
>
> The dual Sierpiński problem: is every Sierpiński number S
> (having a covering set) dual (in the following sense)?
>
> An odd number S is dual
> if and only if for every integer n > 0,
> both S 2^n + 1 and S + 2^n are composite.

For the benefit of those who like me don't have the definition of
"Sierpiński number" cached in our brains:

An odd positive integer S is a Sierpiński number iff no S.2^n+1 is prime.

(So the conjecture here is that this implies that also no 2^n+S is prime.)

> The dual Riesel problem: is every Riesel number R
> (having a covering set) dual (in the following sense)?
>
> An odd number R is dual
> if and only if for every integer n > 0,
> both R 2^n - 1 and |R - 2^n| are composite.

An odd positive integer R is a Riesel number iff no R.2^n-1 is prime.

(So the conjecture here is that this implies that also no 2^n-R is (+-)
prime.)

--
g

[Tomasz Ordowski]

Thanks to Gareth for this additional explanation of the basics.

Let the uninitiated possess the secret of these dual problems.

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[Tomasz Ordowski]

PS. Professor Michael Filaseta (in private correspondence to me) refers to this topic: 

Hi, Thomas,

 

I have not given these problems a lot of thought.  If there is a system of congruences that uses a finite set of primes P to show that k*2^n + 1 is always divisible by a prime in P (how coverings work for this problem), then it is true that k + 2^n will also always be divisible by a prime in P (and vice-versa).  The dual Sierpinski problem (and similarly for the dual Riesel problem) boils down then to whether one can find such a system of congruences for which k*2^n + 1 is not in P for all n but k + 2^n is in P for some n (or vice-versa).  It is reasonable to me to think such a system of congruences exist, but with too many other interesting mathematics I am working on, I don’t see spending time on trying to come up with such a system of congruences.  I might mention this to a student I know who could be interested in the question though.

 

Kind regards,

Michael 

[Tomasz Ordowski]

Professor Filaseta (closing the topic) added:

The dual conjectures are a bit less appealing to me but reasonable.  The difficulty with looking at them from a heuristic point of view is that problems involving covering systems defy heuristics.  One would not think Sierpinski numbers exist based on heuristics, but they do.  Covering systems are basically saying that different divisibility properties can happen to piece together to show the existence of something that defies heuristics.  So I don’t think looking at heuristics for the dual conjectures, which are related to problems on covering systems, makes a lot of sense.