[seqfan] Re: Is it true?

[Trizen]

Dear Tomasz,

I've checked numerically the stronger dual conjectures with the first
several S and R terms, with n <= 2000.

Results for (1):

78557 disproved with n = 31
271129 disproved with n = 197
271577 disproved with n = 5
322523 disproved with n = 1433
327739 disproved with n = 739
482719 disproved with n = 647
575041 disproved with n = 359
603713 holds for odd n < 2000
903983 disproved with n = 37
934909 disproved with n = 5
965431 disproved with n = 13
1259779 disproved with n = 1399
1290677 disproved with n = 19
1518781 disproved with n = 479
1624097 disproved with n = 53
1639459 disproved with n = 7
1777613 disproved with n = 769
2131043 holds for odd n < 2000
2131099 disproved with n = 101
2191531 disproved with n = 317
2510177 disproved with n = 127
2541601 disproved with n = 5
2576089 holds for odd n < 2000
2931767 disproved with n = 11
2931991 disproved with n = 19
3083723 disproved with n = 67
3098059 disproved with n = 59
3555593 disproved with n = 19
3608251 disproved with n = 31

Results for (2):

509203 disproved with n = 431
762701 disproved with n = 11
777149 disproved with n = 2
790841 disproved with n = 2
992077 disproved with n = 3
1106681 disproved with n = 2
1247173 holds for positive n <= 2000
1254341 disproved with n = 2
1330207 disproved with n = 17
1330319 disproved with n = 2
1715053 disproved with n = 13
1730653 disproved with n = 19
1730681 disproved with n = 2
1744117 disproved with n = 23
1830187 disproved with n = 17
1976473 disproved with n = 11
2136283 holds for positive n <= 2000
2251349 disproved with n = 2
2313487 disproved with n = 571
2344211 disproved with n = 11
2554843 disproved with n = 3
2924861 disproved with n = 1489
3079469 disproved with n = 2
3177553 disproved with n = 89
3292241 disproved with n = 37
3419789 disproved with n = 61
3423373 holds for positive n <= 2000
3580901 disproved with n = 5

Best regards,
Daniel

On Tue, Jul 12, 2022 at 7:32 PM Tomasz Ordowski <tomaszordowski at gmail.com>
wrote:

> 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
> >
>
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