[seqfan] Re: Monotonic ordering of nonnegative differences

[Allan Wechsler]

Perhaps sequences like this can be salvaged in the following manner:

Keep the data, and the programs that produced the data. Give the sequence a
primary definition of what the program actually produces (translated into
English). Present the original motivating definition as a conjecture.

On Fri, Oct 11, 2019 at 12:33 PM Max Alekseyev <maxale at gmail.com> wrote:

> Hi Neil,
>
> I'd like to comment on A173671 -- there was "search limit" in my
> submission.
> I now see such a limit in a code submitted by someone else, but this simply
> means that this code may produce incorrect results.
>
> I know two methods of proving that 3^m-2^n=k for a given k is insoluble in
> m,k.
> First is to find a suitable M (if it exists) such that the
> congruence 3^m-2^n == k (mod M) is insoluble (which is easy to verify).
>
> Second is to find the integral points on the following 6 elliptic curves
> corresponding to residues of m and n modulo 2 and 3, respectively:
> y^2 = x^3 + k
> y^2 = 2x^3 + k
> y^2 = 4x^3 + k
> 3y^2 = x^3 + k
> 3y^2 = 2x^3 + k
> 3y^2 = 4x^3 + k
> If in none of the integral points y is a power of 3 and x is a power of 2,
> then 3^m-2^n=k does not have integer solutions in m,n.
> Computing integral points in many cases can done routinely in
> Sage/Magma/etc.
>
> So, I did prove the numbers in my submission A173671, but I cannot say much
> about the later-on additions (e.g., b-file) though.
>
> Regards,
> Max
>
>
> On Fri, Oct 11, 2019 at 11:37 AM Neil Sloane <njasloane at gmail.com> wrote:
>
> > Robert, thank you for catching those errors.  Yes, we will need to add a
> > comment.
> > Sadly, there are b-files too.  Should they be deleted, do you think?
> > Another thing: the complementary sequences are also in the OEIS, e.g.
> > A173671 ,
> > which is the complement of A192111, and was submitted by Max Alekseyev.
> > With a different search limit.  I will handle this, once we decide what
> to
> > do.  Any comments, anyone?
> >
> > We have a rule that programs and b-files should not be based on
> > conjectures, so should the
> > programs be deleted too?
> >
> > I really hope we can keep the sequences, and obviously if we keep the
> > sequences then we need to keep the programs, to show how they were
> > calculated.  But the b-files?
> >
> > 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 Thu, Oct 10, 2019 at 5:31 PM <israel at math.ubc.ca> wrote:
> >
> > > There are 52 sequences from A192110 to A192202, contributed by Clark
> > > Kimberling, with Name of the form "Monotonic ordering of nonnegative
> > > differences a^i-b^j, for i>=0, j>=0" for various values of a and b.
> > >
> > > From the Mathematica code, it seems they are all computed by assuming i
> > <=
> > > 40. I'm not aware of any theoretical justification for the assumption
> > that
> > > any term in the range of the Data (which might go up to several
> million)
> > > will arise from i <= 40, although I have no counterexample and it may
> be
> > > unlikely that there is one. These are related to Catalan's conjecture
> > > (proved by Mihailescu), according to which 1 is not a member of any of
> > > these sequences unless i=1 or j<=1 works. There are also modular
> reasons
> > > for excluding some values (e.g. if prime p divides b but not a, then
> all
> > > terms divisible by p are of the form a^i-1). But for many values > 1, I
> > > don't think much is known rigorously.
> > >
> > > Should these sequences all get a Comment that the Data are conjectured?
> > >
> > > Cheers,
> > > Robert
> > >
> > >
> > > --
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> > >
> >
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> >
>
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