[seqfan] Re: Primes of the form 8n + 7 or -x + 4xy + 4y^2
Neil Sloane
njasloane at gmail.com
Sun Nov 13 21:09:27 CET 2016
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.
Phone: 732 828 6098; home page: http://NeilSloane.com
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.
>> Phone: 732 828 6098; home page: http://NeilSloane.com
>> 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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>>> Seqfan Mailing list - http://list.seqfan.eu/
>>>
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>
>
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