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

[israel at math.ubc.ca]

There are no nontrivial equivalences: the square roots of squarefree 
integers are linearly independent over the rationals. See e.g. 
https://qchu.wordpress.com/2009/07/02/square-roots-have-no-unexpected-linear-relationships/

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) != 
>> > sqrt(13)+sqrt(16)+sqrt(55)+sqrt(67)+sqrt(73)+sqrt(79) you already need 
>> > more than double-precision floating point (their difference is 10^-19, 
>> > see
>> >
>> >
>>  
>>  
>> http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.156.3838&rep=rep1&type=pdf
>> > ).
>> > There is a randomized polynomial time algorithm for testing equality,
>> > but it isn't just compute the result to enough precision.
>> >
>> > Some links with additional information:
>> >
>> >
>>  
>>  
>> https://cstheory.stackexchange.com/questions/79/problems-between-p-and-npc/4010#4010
>> > https://topp.openproblem.net/p33
>> >
>> >
>>  
>>  
>> https://refubium.fu-berlin.de/bitstream/handle/fub188/18449/1993-13.pdf;jsessionid=89F489131478714C98250CFA34AFBFE7?sequence=1
>> >
>> > -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
>> about
>> > 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
>> > >>
>> > >> --
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>> > >>
>> > > --
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>> >
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>> >
>>
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