[seqfan] Re: Monotonic ordering of nonnegative differences

[Neil Sloane]

Allan,  Excellent suggestion! That would certainly be a way to keep the
existing entry pretty much intact.
So A192112, say, would go from
A192112 Monotonic ordering of nonnegative differences 2^i-4^j, for i>=0,
j>=0.
to
A192112 Monotonic ordering of nonnegative differences 2^i-4^j, for
40>=i>=0, j>=0.

(We can't say they are conjectured to be the same, since 2^50 - 1 is in the
former but not in the latter.)

I'm not going to do any more editing of these 52 sequences right now, since
I'm waiting to hear from Max A., who said he would see if he has programs
that could be used to rigorous establish that these sequences are correct
using elliptic curve software. (As he did for A173671)

For the record, here are the 52 sequences:

A192110 <https://oeis.org/A192110>: 2^i-3^j, A192111
<https://oeis.org/A192111>: 3^i-2^j, A192112 <https://oeis.org/A192112>:
2^i-4^j, A192113 <https://oeis.org/A192113>: 4^i-2^j, A192114
<https://oeis.org/A192114>: 2^i-5^j, A192115 <https://oeis.org/A192115>:
5^i-2^j, A192116 <https://oeis.org/A192116>: 2^i-6^j, A192117
<https://oeis.org/A192117>: 6^i-2^j,

A192118 <https://oeis.org/A192118>: 2^i-7^j, A192119
<https://oeis.org/A192119>: 7^i-2^j, A192120 <https://oeis.org/A192120>:
2^i-8^j, A192121 <https://oeis.org/A192121>: 8^i-2^j, A192122
<https://oeis.org/A192122>: 2^i-9^j, A192123 <https://oeis.org/A192123>:
9^i-2^j, A192124 <https://oeis.org/A192124>: 2^i-10^j, A192125
<https://oeis.org/A192125>: 10^i-2^j,

A192147 <https://oeis.org/A192147>: 3^i-4^j, A192148
<https://oeis.org/A192148>: 4^i-3^j, A192149 <https://oeis.org/A192149>:
3^i-5^j, A192150 <https://oeis.org/A192150>: 5^i-3^j, A192151
<https://oeis.org/A192151>: 3^i-6^j, A192152 <https://oeis.org/A192152>:
6^i-3^j, A192153 <https://oeis.org/A192153>: 3^i-7^j, A192154
<https://oeis.org/A192154>: 7^i-3^j,

A192155 <https://oeis.org/A192155>: 3^i-8^j, A192156
<https://oeis.org/A192156>: 8^i-3^j, A192157 <https://oeis.org/A192157>:
3^i-9^j, A192158 <https://oeis.org/A192158>: 9^i-3^j, A192159
<https://oeis.org/A192159>: 3^i-10^j, A192160 <https://oeis.org/A192160>:
10^i-3^j, A192161 <https://oeis.org/A192161>: 4^i-5^j, A192162
<https://oeis.org/A192162>: 5^i-4^j,

A192163 <https://oeis.org/A192163>: 4^i-6^j, A192164
<https://oeis.org/A192164>: 6^i-4^j, A192165 <https://oeis.org/A192165>:
4^i-7^j, A192166 <https://oeis.org/A192166>: 7^i-4^j, A192167
<https://oeis.org/A192167>: 4^i-8^j, A192168 <https://oeis.org/A192168>:
8^i-4^j, A192169 <https://oeis.org/A192169>: 4^i-9^j, A192170
<https://oeis.org/A192170>: 9^i-4^j,

A192171 <https://oeis.org/A192171>: 4^i-10^j, A192172
<https://oeis.org/A192172>: 10^i-4^j, A192193 <https://oeis.org/A192193>:
5^i-6^j, A192194 <https://oeis.org/A192194>: 6^i-5^j, A192195
<https://oeis.org/A192195>: 5^i-7^j, A192196 <https://oeis.org/A192196>:
7^i-5^j, A192197 <https://oeis.org/A192197>: 5^i-8^j, A192198
<https://oeis.org/A192198>: 8^i-5^j,

A192199 <https://oeis.org/A192199>: 5^i-9^j, A192200
<https://oeis.org/A192200>: 9^i-5^j, A192201 <https://oeis.org/A192201>:
5^i-10^j, A192202 <https://oeis.org/A192202>: 10^i-5^j.



and the complements of the first two are A328027 and Max's A173671.


On Sun, Oct 13, 2019 at 4:50 AM Allan Wechsler <acwacw at gmail.com> wrote:

> 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
> > > >
> > > >
> > > > --
> > > > Seqfan Mailing list - http://list.seqfan.eu/
> > > >
> > >
> > > --
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> > >
> >
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