[seqfan] Re: Sequence Idea
David Wilson
davidwwilson at comcast.net
Sat Oct 4 20:54:40 CEST 2014
With all due respect, it seems to me that Neil is confusing the quadratic residues with the multiplicative group.
The elements of the multiplicative group mod n must be nonzero and coprime to the modulus.
The quadratic residues always include 0 and need not be coprime to the modulus.
I respectfully contend that, in the absence of stated or contextual assumptions (which do not exist here) the quadratic residues mod n are by definition the squares mod n.
With regard to gaps, there is also not much leeway for misinterpretation here.
Gaps (as in prime gaps) are the difference between consecutive elements of an ordered set (in this case, a cyclically ordered set).
With respect to a(7) = 3.
Mod 7, the quadratic residues, cyclically ordered, are (..., 0, 1, 2, 4, 0, 1, 2, 4, ....)
Their mod 7 gaps, also cyclically ordered, are (..., 1, 1, 2, 3, 1, 1, 2, 3, ....)
The largest gap is 3.
> -----Original Message-----
> From: SeqFan [mailto:seqfan-bounces at list.seqfan.eu] On Behalf Of Neil
> Sloane
> Sent: Saturday, October 04, 2014 1:39 PM
> To: Sequence Fanatics Discussion list
> Subject: [seqfan] Re: Sequence Idea
>
> The quadratic residues mod 7 are 1,2,4, the nonresidues are 3,5,6.
> Zero is a square mod 7, but isn't usually regarded as a quadratic residue.
>
> There are several questions about A248222.
> 1. How is it defined?
> 2. Why is a(7) = 3 ?
> 3. We ought to have "maximal gap between squares mod n" too.
> 4. There is the question of "consecutive" versus "all" (gap between
> consecutive QR's seems more natural) - in fact, what does "gap" mean?
> 5. If we write the numbers mod 7 in a circle, it is natural to include 0.
> Then we could look at gaps (between QRs, or between squares)
>
> If n is not a prime, there is an even bigger difference between "squares"
> and "QRs". For m to be a QR, we must have gcd(m,n) = 1.
>
> In short, there might be 4 or 8 versions of this sequence
>
> On Sat, Oct 4, 2014 at 12:59 PM, M. F. Hasler <oeis at hasler.fr> wrote:
>
> > David,
> > I agree & propose it as https://oeis.org/draft/A248222.
> > Best,
> > Maximilian
> >
> >
> > On Sat, Oct 4, 2014 at 9:03 AM, David Wilson <davidwwilson at comcast.net>
> > wrote:
> > > Maximum gap between quadratic residues mod n.
> > >
> > >
> > >
> > > Starting at n = 1, I believe a =
> > (1,1,2,3,3,2,3,4,3,3,4,5,5,3,5,7,4,3,5,7,
> > > ...)
> > >
> > >
> > >
> > > I don't think this is in the OEIS.
> > >
> > >
> > > _______________________________________________
> > >
> > > Seqfan Mailing list - http://list.seqfan.eu/
> >
> >
> >
> > --
> > Maximilian
> >
> > _______________________________________________
> >
> > 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
>
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