[seqfan] Re: Repunit wrapped primes

[Neil Sloane]

This is an OK sequence.  Please make enough improvements so that it can be
submitted (and then tell me so I can approve it).

Best regards
Neil

Neil J. A. Sloane, President, OEIS Foundation.
11 South Adelaide Avenue, Highland Park, NJ 08904, USA.
Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ.
Phone: 732 828 6098; home page: http://NeilSloane.com
Email: njasloane at gmail.com



On Sat, Apr 27, 2019 at 11:27 AM Hans L <thehans at gmail.com> wrote:

> Hi,
>
> For p=1823, I have k=9565 as probable prime using BPSW.
>
> Hans Loeblich
>
> On Sat, Apr 27, 2019 at 3:26 AM David Sycamore via SeqFan <
> seqfan at list.seqfan.eu> wrote:
>
> > Dear Seqfans,
> > Let kpk be the number obtained by placing k leading and trailing 1s
> around
> > prime p. Thus, for p=7 and k=1 we get the “repunit wrapped” prime 171,
> > which is composite (3*57)
> >
> > With k=2 we get 11711; also composite (7*1673). However, k=3 gives kpk =
> > 1117111, which is prime, and 3 is the smallest k >0 for which this is the
> > case.
> >
> > For many primes p we can find a k>0 such that kpk is prime, and define a
> > sequence where a(n) is the least k such that kprime(n)k is prime, or -1
> if
> > no such k exists. The sequence is easy to compute in cases where a k
> exists
> > within reach of modest computational means, but there are cases where it
> is
> > not clear that such a k exists or not. The  -1 terms are the challenge
> here.
> >
> > Examples where a(n)=-1:
> >
> > p=2: k2k is composite for all k>0 because it is divisible by the (k+1)-th
> > repunit. Ditto for p=101.
> >
> > p=11: k11k is always divisible by 11 for all k>0.
> >
> > p=37: k37k is composite for all k>0 since its factor cycle comprises a
> > covering congruence for k in which each residue class is associated with
> a
> > particular prime divisor (3, 13 or 37). Therefore any choice of k >0
> leads
> > to a divisor.
> >
> > The trouble starts when p= 397, 563, 739, 1249, 1823...
> >
> > In these cases the factor pattern reveals a specific prime divisor for
> all
> > but one single residue class of the covering congruence of k. For p=397
> we
> > find that k== 1(mod 3)—> 3/kpk, k== 2(mod 3)—>37/kpk, k==3(mod 6)—>7/kpk.
> > But for the remaining k==6 (mod 6) there is always (k<=30000) a prime
> > divisor of k397k,  but it’s never the same one, so there is apparently no
> > possibility to predict a divisor for any k divisible by 6.  Therefore all
> > we can say is that if there is a k that makes k397k prime, then it must
> be
> > a multiple of 6 and >30000. Alternatively a proof is needed that for all
> > k==6 (mod 6) k397k is composite.
> >
> > To make matters worse,  there are cases like the above (a residue class
> > with no distinct prime divisor associated with it) for which we do
> > eventually reach a k such that  kpk becomes prime. An example of this is
> > p=61, for which k=repunit of length 14 gives k61k prime, and 14 belongs
> to
> > k=={0,2,5} (mod 6) which has no fixed prime divisor associated with it,
> > whereas the other two, k==1 (mod2) and k== {3,4} (mod 6) are associated
> > with prime divisors 3 and 13 respectively. There are plenty of other
> > examples like this one.
> >
> > It’s a bit like the de Polignac problem, and whilst the use of covering
> > congruences is promising, there are cases where It is hard to be sure
> for a
> > given p, whether a k>0 exists or not.
> >
> > Can anyone offer suggestions to resolve this question for the troublesome
> > primes mentioned above?
> >
> > I have entered draft sequence A306861 in an attempt to record  the least
> k
> > for which kpk is prime,  and -1 if no such k exists. The current draft,
> > including data up to n=86 includes prime(78)=397, for which an uncertain
> > (but perhaps highly likely) term -1 has been recorded.
> > The other four “-1” terms in the present data (p=2,11,37,101) can be
> > confirmed as explained above. Assuming the sequence is accepted there
> will
> > be a need to confirm values for n> 86, which would include the
> > indeterminate cases mentioned above (eventually for a b-file).
> >
> > Included in the Comments of draft A306861 is a conjecture that there are
> > an infinite number of -1 terms (primes p for which kpk is composite for
> all
> > k>0).  Can anyone suggest a proof, or argue that the list of such primes
> is
> > finite?
> >
> > I would like to improve draft A306861 and submit it for review sometime
> > soon.  Any assistance gratefully received.
> >
> > Best regards
> >
> > David.
> >
> >
> >
> >
> >
> >
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> >
>
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