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

jean-paul allouche jean-paul.allouche at imj-prg.fr
Sun Nov 13 20:52:02 CET 2016

```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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```