# [seqfan] Re: Sequence Idea

Neil Sloane njasloane at gmail.com
Sat Oct 4 21:08:34 CEST 2014

```David said:
> The quadratic residues always include 0 and need not be coprime to the
modulus.
This is not the usual definition, which is the following:

Ireland and Rosen, page 50: "If (a,m)=1, a is called a quad res mod m if
the congruence x^2 == a (mod m) has a solution. Otherwise a is called a

This excludes 0 and does require relatively prime.

It is pointless to argue about definitions.  The OEIS should contain all
possible interpretations of this notion.

On Sat, Oct 4, 2014 at 2:54 PM, David Wilson <davidwwilson at comcast.net>
wrote:

> 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.
> > Email: njasloane at gmail.com
> >
> > _______________________________________________
> >
> > 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.