[seqfan] Re: Worry about old sequence, A030077, paths in K_n

Brendan McKay Brendan.McKay at anu.edu.au
Mon Apr 4 04:04:16 CEST 2022

```It isn't about rational combinations of square roots of integers, and
square roots don't even need to be involved.  The problem is to test a
sum for equality to 0 where each term is an integer times sin(k pi / n)
for integer k.  Robert Gerbicz gave a correct analysis and deterministic
algorithm before, and I noted that Maple can do it very quickly in all
cases I tried (thousands).

Brendan.

On 4/4/2022 5:16 am, israel at math.ubc.ca wrote:
> There are no nontrivial equivalences: the square roots of squarefree
> integers are linearly independent over the rationals. See e.g.
>
> Cheers,
> Robert
>
> On Apr 3 2022, Allan Wechsler wrote:
>
>> Are there any good examples of nontrivial equivalences? I'm thinking
>> that
>> we might be able to *characterize* nontrivial equivalences, and thus be
>> able to prove two paths to be unequal in a sort of "combinatorial" way,
>> without resorting to extended arithmetic.
>>
>> On Sat, Apr 2, 2022 at 3:03 PM Sean A. Irvine <sairvin at gmail.com> wrote:
>>
>>> By the way, my code is using computable reals not floating-point,
>>> but it still faces the problem of deciding equality. I've been using
>>> 50 decimal digits for this problem. I could easily rerun with higher
>>> precision, but I would like to remove the need for the approximation
>>> altogether.
>>>
>>> Sean.
>>>
>>>
>>> On Sun, 3 Apr 2022 at 02:04, David Applegate <david at bcda.us> wrote:
>>>
>>> > I worry about using floating point (extended or not) to check if >
>>> different sums of square roots are equal or not. Using finite >
>>> precision for this is extremely tricky This is a notoriously hard >
>>> problem in general. For example, to see that >
>>> sqrt(7)+sqrt(14)+sqrt(39)+sqrt(70)+sqrt(72)+sqrt(76)+sqrt(85) != >
>>> need > more than double-precision floating point (their difference
>>> is 10^-19, > see
>>> >
>>> >
>>>
>>>
>>>
>>> > ).
>>> > There is a randomized polynomial time algorithm for testing equality,
>>> > but it isn't just compute the result to enough precision.
>>> >
>>> >
>>> >
>>>
>>>
>>>
>>> >
>>> >
>>> >
>>>
>>>
>>>
>>> >
>>> > -Dave
>>> >
>>> > On 4/2/2022 5:05 AM, Sean A. Irvine wrote:
>>> > > So far I can verify to A030077(14):
>>> > >
>>> > > 2022-04-02 21:46:54 1
>>> > > 2022-04-02 21:46:54 1
>>> > > 2022-04-02 21:46:54 1
>>> > > 2022-04-02 21:46:54 3
>>> > > 2022-04-02 21:46:54 5
>>> > > 2022-04-02 21:46:54 17
>>> > > 2022-04-02 21:46:54 28
>>> > > 2022-04-02 21:46:54 105
>>> > > 2022-04-02 21:46:54 161
>>> > > 2022-04-02 21:46:54 670
>>> > > 2022-04-02 21:46:55 1001
>>> > > 2022-04-02 21:47:00 2869
>>> > > 2022-04-02 21:47:58 6188
>>> > > 2022-04-02 22:01:58 26565
>>> > >
>>> > > I adapted my existing program for A007874 to this case.  I'm not
>>> sure
>>> > why I
>>> > > skipped over it before, but perhaps the dihedral group confused me.
>>> The
>>> > > program uses 50 digits of precision for the length determination. I
>>> feel
>>> > > like it should be possible to do this without any kind of precision
>>> > limit,
>>> > > but I don't have the time for that now.  I'll leave it running
>>> overnight.
>>> > >
>>> > > I'll also start a run for A007874 itself which also has four > >
>>> additional terms with a(15) looking a little suspicious.
>>> > >
>>> > > Sean.
>>> > >
>>> > >
>>> > >
>>> > > On Sat, 2 Apr 2022 at 16:22, Neil Sloane <njasloane at gmail.com>
>>> wrote:
>>> > >
>>> > >> To Seq Fans, The creator of an interesting sequence, A030077, >
>>> >> Daniel Gittelson, submitted 12 terms in 1999, and 4 more terms
>>> were > >> added in
>>> > 2007
>>> > >> by a former editor. Daniel G. wrote to me today, expressing doubt
>>> > the
>>> > >> 4 additional terms.
>>> > >>
>>> > >> He says he interrupted his study of sequences to pursue a medical
>>> > career,
>>> > >> but now that he is retired, he can return to combinatorics.
>>> > >>
>>> > >> It would be nice if someone could verify the terms! (It is not
>>> at > >> all obvious to me how to do this.)
>>> > >>
>>> > >> Neil Sloane
>>> > >>
>>> > >> --
>>> > >> Seqfan Mailing list -
>>> > >>
>>> > > --
>>> > > Seqfan Mailing list -
>>> >
>>> > --
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>>>
>>> --
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>>
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