# [seqfan] Re: Primes of the form 8n + 7 or -x + 4xy + 4y^2

Alonso Del Arte alonso.delarte at gmail.com
Sun Nov 13 21:14:58 CET 2016

```Neil, you're absolutely right. What I've said so far does not guarantee
that for every n there are x and y that will give the same value as 8n + 7
in -x^2 + 4xy + 4y^2. So that's what I'm going to be thinking about now.

On Sun, Nov 13, 2016 at 3:09 PM, Neil Sloane <njasloane at gmail.com> wrote:

> Al,  You said:
>
> If y is even, then 4xy + 4y^2 is a multiple of 8. With y even and x odd, we
> see that x^2 = 1 mod 8, which "flips over" to -x^2 = 7 mod 8. Therefore,
> primes of the form -x + 4xy + 4y^2 are a trivial, not proper, subset of
> primes of the form 8n + 7.
>
> I guess it still remains to show that every prime of the form 8n+7 can be
> reached in this way?
>
> Best regards
> Neil
>
> 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
>
>
> On Sun, Nov 13, 2016 at 2:52 PM, jean-paul allouche <
> jean-paul.allouche at imj-prg.fr> wrote:
>
> > Hi
> >
> > There is a remark in A141175
> > >>>>>
> > Values of the quadratic form are {0,4,7} mod 8, so this is a subset of
> > A007522. - R. J. Mathar, Jul 30 2008
> > >>>>>
> > but you say that this is the whole set, not only a subset, right?
> > (by the way there is a typo in the mail: it isn't -x + 4xy +4y^2
> > but -x^2 + 4xy + 4y^2)
> >
> > best
> > jp allouche
> >
> >
> >
> > Le 13/11/16 à 20:33, Neil Sloane a écrit :
> >
> > Al,  I agree with your reasoning, and I will edit the two sequences as
> you
> >> suggest.
> >>
> >> Best regards
> >> Neil
> >>
> >> 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
> >>
> >>
> >> On Sun, Nov 13, 2016 at 2:23 PM, Alonso Del Arte <
> >> alonso.delarte at gmail.com>
> >> wrote:
> >>
> >> I am now convinced that A007522, primes of the form 8n + 7, and A141175,
> >>> primes of the form -x + 4xy + 4y^2, are indeed the same sequence.
> >>>
> >>> If both x and y are even, then it doesn't matter, -x + 4xy + 4y^2 gives
> >>> us
> >>> a composite even number. If x is even but y is odd then the formula
> still
> >>> gives us a composite even number.
> >>>
> >>> If y is even, then 4xy + 4y^2 is a multiple of 8. With y even and x
> odd,
> >>> we
> >>> see that x^2 = 1 mod 8, which "flips over" to -x^2 = 7 mod 8.
> Therefore,
> >>> primes of the form -x + 4xy + 4y^2 are a trivial, not proper, subset of
> >>> primes of the form 8n + 7.
> >>>
> >>> This is correct and complete, right? Given that A141175 has been in the
> >>> OEIS for almost a decade, it should be labeled a duplicate and have its
> >>> relevant information not already in A007522 copied over to the older
> >>> entry?
> >>>
> >>> Al
> >>>
> >>> --
> >>> Alonso del Arte
> >>> Author at SmashWords.com
> >>> <https://www.smashwords.com/profile/view/AlonsoDelarte>
> >>> Musician at ReverbNation.com <http://www.reverbnation.com/
> alonsodelarte>
> >>>
> >>> --
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> >>>
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> >>
> >
> >
> > --
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> >
>
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>

--
Alonso del Arte
Author at SmashWords.com
<https://www.smashwords.com/profile/view/AlonsoDelarte>
Musician at ReverbNation.com <http://www.reverbnation.com/alonsodelarte>

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