[seqfan] Re: Monotonic ordering of nonnegative differences

[Max Alekseyev]

Correction: I meant to say - there was NO "search limit" in my submission.

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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>