[seqfan] Re: Monotonic ordering of nonnegative differences

[Max Alekseyev]

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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