[seqfan] Re: Is it true?

[Tomasz Ordowski]

Dear Professor Michael,

I am grateful for your comprehensive reply to the first part of my letter.

To encourage you to rethink the second part,
I have formulated two stronger dual conjectures:

(1) There are Sierpinski numbers S such that
both ((2S)^n+1)/(2S+1) and (S^n+2^n)/(S+2) are composite for every odd n >
1.

(2) There are Riesel numbers R such that
both ((2R)^n-1)/(2R-1) and (R^n-2^n)/(R-2) are composite for every n > 1.

Please think about it in your spare time.

Have a nice summer vacation!

Sincerely,

Tom

P.S. Is there any chance to prove the Sierpinski or Riesel dual conjectures?


wt., 12 lip 2022 o 10:02 Filaseta, Michael <filaseta at math.sc.edu>
napisał(a):

> Dear Tomasz,
>
>
>
> I am not sure what your "uncertain source" is, but I doubt that (*) and
> (**) are true.  In fact,
>
> 1694916068074761354577
>
> is a Sierpinski number and
>
> 1694916068074761354577^5*2^142+1
>
> appears to be a prime, so
>
> 1694916068074761354577^5
>
> is not a Sierpinski number.
>
>
>
> However, what Arkadiusz Wesolowski wrote in the The On-Line Encyclopedia
> of Integer Sequences is correct.  He wrote that 78557^p is a Sierpinski
> number for every prime p > 3.  There is something more general, but it is
> not what you wrote in (*).  If k is a Sierpinski number that is derived
> from a covering system in the usual way with moduli from a finite set M and
> t is relatively prime to the lcm of the moduli in M, then k^t is a
> Sierpinski number.  Many of the small Sierpinski numbers, like 78557,
> come from using a set M where the lcm of the elements of M is only
> divisible by the primes 2 and 3.  For those then k^p is a Sierpinski
> number for every p > 3 (and k^t as well as long as gcd(t,6) = 1).  The
> example above comes from a covering with moduli having lcm that is
> divisible by 2, 3 and 5.  So one would need p > 5 in this case.
>
>
>
> Note that (*) is also not likely true if a Sierpinski number does not come
> from a covering system, which as noted in the links you gave likely exists.
>
>
>
> The analogous comments apply to Riesel numbers.  There is no reason to
> believe (**) but it p > 3 is replaced by p sufficiently large (depending on
> the Riesel number) or a p that does not divide any of the moduli in the
> covering system to create the Riesel number, then like (*) the result holds.
>
>
>
> Arkadiusz Wesolowski is also quoted in The On-Line Encyclopedia of Integer
> Sequences as saying, "a(1) = 78557 is also the smallest odd n for which
> either n^p*2^k + 1 or n^p + 2^k is composite for every k > 0 and every
> prime p greater than 3."  He may have meant, "a(1) = 78557 is also the
> smallest odd n for which both n^p*2^k + 1 and n^p + 2^k are composite for
> every k > 0 and every prime p greater than 3."  I verified this statement
> but not the statement he made.  To clarify, the difficulty has to do with
> what numbers are not yet resolved from the 17 or Bust project that are <
> 78557 and possibly Sierpinski numbers.  So one needs to check these.  For
> example, we do not know yet if n = 21181 is a Sierpinski number.  But one
> can check that in this case n^5 + 2^188 is a prime, so even if n = 21181 is
> a Sierpinski number, it does not satisfy that both n^p*2^k + 1 and n^p +
> 2^k are composite for every k > 0 and every prime p greater than 3.
>
>
>
> As to (1) and (2), I am unfortunately busy and do not have time to think
> about these or to write up arguments for the statements above.  It may
> however be relatively easy to answer these by looking at explicit
> examples (like 78557) in detail.
>
>
>
> Hopefully, these clarifications help.
>
>
>
> Kind regards,
>
> Michael
>
>
>
>
>
> *From: *Tomasz Ordowski <tomaszordowski at gmail.com>
> *Date: *Monday, July 11, 2022 at 10:47 AM
> *To: *Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>
> *Cc: *Filaseta, Michael <filaseta at math.sc.edu>
> *Subject: *Is it true?
>
>  Dear readers,
>
>
>
> I found out from an uncertain source that:
>
> (*) If S is a Sierpinski number, then S^p is also a Sierpinski number for
> every prime p > 3.
>
> (**) If R is a Riesel number, then R^p is also a Riesel number for every
> prime p > 3.
>
> Cf. Wesolowski's comments on A076336 * and A101036 **.
>
> See A076336 - OEIS
> <https://nam02.safelinks.protection.outlook.com/?url=http%3A%2F%2Foeis.org%2FA076336&data=05%7C01%7Cfilaseta%40math.sc.edu%7C02cb10db4ead4c924cb008da634c4162%7C4b2a4b19d135420e8bb2b1cd238998cc%7C0%7C0%7C637931476517638175%7CUnknown%7CTWFpbGZsb3d8eyJWIjoiMC4wLjAwMDAiLCJQIjoiV2luMzIiLCJBTiI6Ik1haWwiLCJXVCI6Mn0%3D%7C2000%7C%7C%7C&sdata=qXUYCf5QrGBZFbDNMtYxBL7r3R9S3Fs9SpHLY9yLsGk%3D&reserved=0>
>  and A101036 - OEIS
> <https://nam02.safelinks.protection.outlook.com/?url=http%3A%2F%2Foeis.org%2FA101036&data=05%7C01%7Cfilaseta%40math.sc.edu%7C02cb10db4ead4c924cb008da634c4162%7C4b2a4b19d135420e8bb2b1cd238998cc%7C0%7C0%7C637931476517638175%7CUnknown%7CTWFpbGZsb3d8eyJWIjoiMC4wLjAwMDAiLCJQIjoiV2luMzIiLCJBTiI6Ik1haWwiLCJXVCI6Mn0%3D%7C2000%7C%7C%7C&sdata=jN7%2B0CuyFLijAue4Usn5ZbIubqICMoFWJO%2BHNYU2zL4%3D&reserved=0>
>
> I am asking for better references.
>
>
>
> Therefore, the following conjectures can be put forward:
>
> (1) There are Sierpinski numbers S such that ((2S)^n+1)/(2S+1) is
> composite for every odd n > 3.
>
> (2) There are Riesel numbers R such that ((2R)^n-1)/(2R-1) is composite
> for every n > 3.
>
> Note that the above formulas can give primes only for prime numbers n.
>
> Which candidates are easy to eliminate numerically?
>
> And which of the rest are provable?
>
>
>
> Best regards,
>
>
>
> Thomas Ordowski
>