[seqfan] Re: a^2 + b^3 = c^4

[Alex M]

But then we face the question of -- what if someone wanted to find just the
values of a? We could sort by values of a, or values of b, or values of c;
we could have duplicates, or not, for the sequence we're sorting by. This
gives me 12 sequences. It quickly becomes unreasonable. :-/

~6 out of 5 statisticians say that the
number of statistics that either make
no sense or use ridiculous timescales
at all has dropped over 164% in the
last 5.62474396842 years.


On Tue, May 6, 2014 at 3:16 PM, Charles Greathouse <
charles.greathouse at case.edu> wrote:

> You mean there would have to be 6 sequences? I agree.
>
> Charles Greathouse
> Analyst/Programmer
> Case Western Reserve University
>
>
> On Tue, May 6, 2014 at 5:26 PM, Andrew N W Hone <A.N.W.Hone at kent.ac.uk
> >wrote:
>
> > The problem is that the sequences will not make any sense without
> repeated
> > numbers in the c sequence.
> >
> > 72 appears twice in the cc sequence because (as far as I understand what
> > has been computed) there are
> > precisely two pairs of positive integers (a,b) such that a^2+b^3 = c^4.
> As
> > we continue increasing c,
> > it is likely that a particular value will need to be repeated more than
> > twice (arbitrarily many times?) - as
> > many times as there are corresponding pairs of values of a and b.
> >
> > If the c values are not repeated, then the (a,b) values will soon be "out
> > of sync" with the values of c:
> > e.g. eventually it might happen that the 150th terms in the a/b sequences
> > correspond to the 145th value of c,
> > and so on, with the  gap getting increasingly wider.
> >
> > Andy
> > ________________________________________
> > From: SeqFan [seqfan-bounces at list.seqfan.eu] on behalf of Charles
> > Greathouse [charles.greathouse at case.edu]
> > Sent: 06 May 2014 16:13
> > To: Sequence Fanatics Discussion list
> > Subject: [seqfan] Re: a^2 + b^3 = c^4
> >
> > We may want to add those ancillary sequences and link them to the main
> > sequence, sure.
> >
> > Charles Greathouse
> > Analyst/Programmer
> > Case Western Reserve University
> >
> >
> > On Tue, May 6, 2014 at 10:02 AM, Lars Blomberg <lars.blomberg at visit.se
> > >wrote:
> >
> > > As I understand it, one of the main uses of OEIS it to find a possible
> > > sequence based on a few values that has been found in some
> investigation.
> > >
> > > Say we have found 71,72,75 and look for it,  then we will not find the
> > > proposed "cc" sequence.
> > > We have no way of knowing that 72 should be entered twice.
> > > Similarly looking for 108,126,128 will not find the "bb" sequence
> because
> > > the order is not the same.
> > >
> > > Would it not be better to let the "c" sequence have the c values in
> order
> > > without duplicates.
> > > The values a,b,c for all the solutions (including duplicates) can be
> > > supplied as a file.
> > >
> > > The same for b (and a if we can compute it).
> > >
> > > /Lars
> > >
> > > -----Ursprungligt meddelande----- From: Jean-François Alcover
> > > Sent: Tuesday, May 06, 2014 3:10 PM
> > >
> > > To: Sequence Fanatics Discussion list
> > > Subject: [seqfan] Re: a^2 + b^3 = c^4
> > >
> > > The 3 "co-ordinated" sequences might look like this:
> > >
> > > aa = {28, 27, 63, 1176, 648, 433, 1792, 2925, 3807, 4785, 4941, 1728,
> > 4500,
> > > 6083, 7452, 7203,...}
> > >
> > > bb = {8, 18, 36, 49, 108, 143, 128, 126, 108, 136, 135, 288, 225, 23,
> > 216,
> > > 343,...}
> > > cc = {6, 9, 15, 35, 36, 42, 48, 57, 63, 71, 72, 72, 75, 78, 90, 98,...}
> > >
> > > jfa
> > >
> > > 2014-05-06 12:18 GMT+02:00 Andrew N W Hone <A.N.W.Hone at kent.ac.uk>:
> > >
> > >  I'm not sure how the c sequence works as an index on the other two,
> > since
> > >> for the same
> > >> value of c there could be more than one pair (a,b) which works.
> > >>
> > >> For fixed c, this is an elliptic curve in the (a,b) plane. It is more
> > >> commonly written as
> > >>
> > >> y^2 = x^3 + d
> > >>
> > >> taking (x,y,d) = (-b, a, c^4). Siegel's theorem says that for fixed d
> > >> (i.e. fixed c) there are
> > >> only finitely many integer solutions.
> > >>
> > >> For a beautiful introduction to Siegel's theorem see the article
> > >>
> > >> http://arxiv.org/pdf/1005.0315v3.pdf
> > >>
> > >> which is published in American Mathematical Monthly.
> > >>
> > >> All the best
> > >> Andy
> > >> ________________________________________
> > >> From: SeqFan [seqfan-bounces at list.seqfan.eu] on behalf of Charles
> > >> Greathouse [charles.greathouse at case.edu]
> > >> Sent: 05 May 2014 15:50
> > >> To: Sequence Fanatics Discussion list
> > >> Subject: [seqfan] Re: a^2 + b^3 = c^4
> > >>
> > >> That's the way I would do it -- add c, then a and b sequences indexed
> on
> > >> c.
> > >>
> > >> Charles Greathouse
> > >> Analyst/Programmer
> > >> Case Western Reserve University
> > >>
> > >>
> > >> On Mon, May 5, 2014 at 10:38 AM, Jean-François Alcover <
> > >> jf.alcover at gmail.com
> > >> > wrote:
> > >>
> > >> > I agree: the c-sequence should be the first, and, in my opinion,
> > >> > if should serve as index to the a- and b-sequence,
> > >> > and show the duplicates this way:
> > >> > 6, 9, 15, 35, 36, 42, 48, 57, 63, 71, 72, 72, 75, 78, 90, 98, 100,
> > >> > 100, 120, 135, 141, 147, 147, 162, 195, 196, 204, 208, 215,
> > >> > 225, 225, 225, 243, ...
> > >> >
> > >> > jfa
> > >> >
> > >> > 2014-05-05 16:08 GMT+02:00 Alonso Del Arte <
> alonso.delarte at gmail.com
> > >:
> > >> >
> > >> > > I would add the "c" sequence first, and hold off on "a" and "b"
> > until
> > >> > > I
> > >> > or
> > >> > > someone else can resolve the theoretical questions, like which
> which
> > >> > values
> > >> > > of c have more than one pair of a and b, and whether there is
> such a
> > >> > thing
> > >> > > as a "primitive" solution.
> > >> > >
> > >> > > Al
> > >> > >
> > >> > >
> > >> > > On Mon, May 5, 2014 at 3:00 AM, Lars Blomberg <
> > lars.blomberg at visit.se
> > >> > > >wrote:
> > >> > >
> > >> > > > Putting each of a,b,c in increasing order is the logical thing
> to
> > >> do, I
> > >> > > > agree.
> > >> > > >
> > >> > > > I have computed some c values and my (somewhat hasty) thought
> was
> > to
> > >> > > > include the
> > >> > > > corresponding a and b values as separate sequences.
> > >> > > > But as you point out, this will not be correct.
> > >> > > >
> > >> > > > Maybe I will stick with the "c" sequence for the time being.
> > >> > > >
> > >> > > > /Lars
> > >> > > >
> > >> > > > -----Ursprungligt meddelande----- From: israel at math.ubc.ca
> > >> > > > Sent: Monday, May 05, 2014 8:35 AM
> > >> > > > To: Sequence Fanatics Discussion list
> > >> > > > Subject: [seqfan] Re: a^2 + b^3 = c^4
> > >> > > >
> > >> > > >
> > >> > > > What order would you put these in?
> > >> > > > It would seem logical to put them each in increasing order.
> > >> > > > Thus the "a" sequence would be the set of all a such that
> > >> > > > a^2 + b^3 = c^4 for some b and c.
> > >> > > > However, searching for solutions may be difficult: I don't know
> > >> > > > if there are effective bounds on b and c for given a.
> > >> > > > The other two should be OK: for the "c" sequence we certainly
> have
> > >> > > > a < c^2 and b < c^(4/3), while for the "b" sequence, since
> > >> > > > b^3 = (c^2+a)(c^2-a) > c^2 + a, so c < b^(3/2) and a < b^3.
> > >> > > >
> > >> > > > Robert Israel
> > >> > > > University of British Columbia and D-Wave Systems
> > >> > > >
> > >> > > > On May 4 2014, Lars Blomberg wrote:
> > >> > > >
> > >> > > >  Hello Seqfans,
> > >> > > >>
> > >> > > >> a^2 + b^3 = c^4 has solutions
> > >> > > >> a = 28, 27, 63, 1176, 648, 433, 1792, ...
> > >> > > >> b = 8, 18, 36, 49, 108, 143, 128, ...
> > >> > > >> c = 6, 9, 15, 35, 36, 42, 48, ...
> > >> > > >> none of which seem to be in OEIS.
> > >> > > >>
> > >> > > >> I intend to add the "c" sequence.
> > >> > > >> One question though: Should I add the "a" and "b" sequences as
> >
> > >
> > >> >> well?
> > >> > > >>
> > >> > > >> /Lars
> > >> > > >>
> > >> > > >> _______________________________________________
> > >> > > >>
> > >> > > >> Seqfan Mailing list - http://list.seqfan.eu/
> > >> > > >>
> > >> > > >>
> > >> > > >>
> > >> > > > _______________________________________________
> > >> > > >
> > >> > > > Seqfan Mailing list - http://list.seqfan.eu/
> > >> > > >
> > >> > > > _______________________________________________
> > >> > > >
> > >> > > > Seqfan Mailing list - http://list.seqfan.eu/
> > >> > > >
> > >> > >
> > >> > >
> > >> > >
> > >> > > --
> > >> > > Alonso del Arte
> > >> > > Author at SmashWords.com<
> > >> > > https://www.smashwords.com/profile/view/AlonsoDelarte>
> > >> > > Musician at ReverbNation.com <
> > >> http://www.reverbnation.com/alonsodelarte>
> > >> > >
> > >> > > _______________________________________________
> > >> > >
> > >> > > Seqfan Mailing list - http://list.seqfan.eu/
> > >> > >
> > >> >
> > >> > _______________________________________________
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> > >> > Seqfan Mailing list - http://list.seqfan.eu/
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