[seqfan] Re: Numbers and 'forms'.

[Neil Sloane]

Continuing to respond to Peter's question about the meaning of "represent":
I've written a lot of papers about quadratic forms (see for example Chapter
15 of Sphere Packings, Lattices and Groups), and I guess I own most of the
books on the subject. The naive meaning
of "represent" is universally adopted in the literature on quadratic forms
(another reference:  Gerstein, Basic Quad. Forms, AMS Grad. Studies in
Math, Vol 90, page 2).

Of course there is nothing to stop someone from saying that by
represent, they mean represent nontrivially - not allowing x=y=0 -
but in the OEIS we will continue to use the naive meaning, and add
"nontrivially" when that is appropriate.

Neil


On Fri, Jun 27, 2014 at 10:15 PM, Neil Sloane <njasloane at gmail.com> wrote:

> Well, in the OEIS we follow Buell, Cox, et al., and use the obvious
> meaning of "represent". I know Andrew well, and in fact I was on his thesis
> committee.
>
> I've just about finished my
> survey of seqs arising from binary quadratic forms - see next message.
>
> There is lots of work that needs doing.
>
> Neil
>
>
> On Fri, Jun 27, 2014 at 7:44 PM, Charles Greathouse <
> charles.greathouse at case.edu> wrote:
>
>> It seems clear that multiple definitions are in use. The best thing we can
>> do is be explicit in the individual sequences so neither camp will be
>> confused.
>>
>> Charles Greathouse
>> Analyst/Programmer
>> Case Western Reserve University
>>
>>
>> On Fri, Jun 27, 2014 at 6:31 PM, Peter Luschny <peter.luschny at gmail.com>
>> wrote:
>>
>> > NJAS> Peter said " a form represents n by
>> > NJAS> integers x, y if and only if x != 0."
>> >
>> > Yes, I said it, but I did not invent this definition.
>> >
>> > NJAS> The authorities disagree.
>> > NJAS> [...] David Cox, "Primes of the Form x^2+ny^2"
>> > NJAS> [...] Buell, Binary Quadratic Forms
>> >
>> > I took the definition from Andrew Sutherland [1] who is
>> > Principal Research Scientist in Computational Number Theory
>> > at the MIT and was recently awarded the Selfridge Prize
>> > (by the way like John Voight).
>> >
>> > So at least it is worth to listen to his arguments.
>> >
>> > To see Sutherland's arguments in place look at his
>> > lectures "Introduction to Arithmetic Geometry" [2] and [3]
>> > which were chosen as MIT course ware. The relevant
>> > chapter is [4]. See definition 9.7, example 9.8 and
>> > the comment following it.
>> >
>> > Perhaps I am missing something and I am not an expert
>> > who can evaluate diligently the relative merits of the
>> > two definitions but judging from the context in Sutherland's
>> > lecture I am inclined to follow his definition.
>> >
>> > Peter
>> >
>> > [1] https://math.mit.edu/people/profile.php?pid=272
>> > [2]
>> >
>> http://ocw.mit.edu/courses/mathematics/18-782-introduction-to-arithmetic-geometry-fall-2013/
>> > [3]
>> >
>> http://ocw.mit.edu/courses/mathematics/18-782-introduction-to-arithmetic-geometry-fall-2013/lecture-notes/
>> > [4]
>> >
>> http://ocw.mit.edu/courses/mathematics/18-782-introduction-to-arithmetic-geometry-fall-2013/lecture-notes/MIT18_782F13_lec9.pdf
>> >
>> > _______________________________________________
>> >
>> > Seqfan Mailing list - http://list.seqfan.eu/
>> >
>>
>> _______________________________________________
>>
>> Seqfan Mailing list - http://list.seqfan.eu/
>>
>
>
>
> --
> Dear Friends, I have now retired from AT&T. New coordinates:
>
> 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
>
>


-- 
Dear Friends, I have now retired from AT&T. New coordinates:

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