# [seqfan] Re: Another duplicate: primes of the form x^2 + 4xy - 4y^2

Sun Jan 29 10:04:49 CET 2017

```Dear Jan,

Before, for every positive integer y I found an odd only integer x>0,
if p=8n+1. This solves the problem. (By formula x=-2y+sqrt(p+8y^2)
two such distinct x's cannot correspond to the same y).
Moreover, I proved that if p=8n+1, then n has the form
n=k^2-k+(i^2-i)/2+2ki.

Best regards,
________________________________________
From: SeqFan [seqfan-bounces at list.seqfan.eu] on behalf of Jan Orwat [johnorwhat at gmail.com]
Sent: 28 January 2017 17:36
To: Sequence Fanatics Discussion list
Subject: [seqfan] Re: Another duplicate: primes of the form x^2 + 4xy - 4y^2

Since 8y^2+p has to be a perfect square, p should be of the form u^2-8y^2.
On the other hand we have A038873 containing primes congruent to {1, 2, 7}
mod 8, also being of the form u^2-2v^2. If u is even, we have 2, if u is
odd, then v odd may give only primes congruent to 7 mod 8, and  v even may
give p congruent to 1 modulo 8. Let's write v = 2y, and we get form u^2 -
8y^2. Since every prime congruent to {1, 2, 7} mod 8 is of the form
u^2-2v^2 (as stated in A038873 comments), every prime congruent to 1 mod 8
has to be of the form u^2-8y^2. Therefore A141174 and A007519 are
equivalent. QED
Hope it is correct reasoning.

Best regards,
Jan

2017-01-28 4:52 GMT+01:00 Alonso Del Arte <alonso.delarte at gmail.com>:

>
> Thank you very much. I'm starting to work on editing A141174 accordingly.
>
> Al
>
> On Thu, Jan 26, 2017 at 5:08 PM, Vladimir Shevelev <shevelev at bgu.ac.il>
> wrote:
>
> > Dear Al,
> >
> > A necessary and sufficient condition of representation of p=8n+1
> > {8y^2+8n+1 is perfect square}, since only in this case
> > solving square equation for x, we have
> > x=-2y+sqrt(8y^2+8n+1) is integer.
> > For this  a sufficient condition is { n has a form
> > n=k^2-k + i(4k+i-1)/2, i>=0, k>=1}.
> >  In this case  x=2i + 2k-1. y=k.
> >
> > Best regards,
> >
> >
> >
> > ________________________________________
> > From: SeqFan [seqfan-bounces at list.seqfan.eu] on behalf of Alonso Del
> Arte
> > [alonso.delarte at gmail.com]
> > Sent: 25 January 2017 19:15
> > To: Sequence Fanatics Discussion list
> > Subject: [seqfan] Another duplicate: primes of the form x^2 + 4xy - 4y^2
> >
> > run it by other people before going ahead with it.
> >
> > A141174, primes of the form x^2 + 4xy - 4y^2, is a duplicate of A007519,
> > primes of the form 8n + 1.
> >
> > I've already done the easy step, proving that all primes of the form x^2
> +
> > 4xy - 4y^2 are congruent to 1 mod 8. Since x^2 + 4xy - 4y^2 = 2 or -2 is
> > impossible, x must be odd. And since x is odd, x^2 = 1 mod 8.
> >
> > If y is even, then both 4xy and 4y^2 are multiples of 8. If y is odd,
> then
> > 4xy = 4 mod 8, but so is 4y^2, cancelling out the effect and leaving x^2
> =
> > 1 mod 8.
> >
> > There's still the issue of proving every prime of the form 8n + 1 has an
> > x^2 + 4xy - 4y^2 representation. With the previous duplicate, this was
> > proven with quadratic forms, if I recall correctly.
> >
> > Al
> >
> > --
> > Alonso del Arte
> > Author at SmashWords.com
> > <https://www.smashwords.com/profile/view/AlonsoDelarte>
> > Musician at ReverbNation.com <http://www.reverbnation.com/alonsodelarte>
> >
> > --
> > Seqfan Mailing list - http://list.seqfan.eu/
> >
> > --
> > Seqfan Mailing list - http://list.seqfan.eu/
> >
>
>
>
> --
> Alonso del Arte
> Author at SmashWords.com
> <https://www.smashwords.com/profile/view/AlonsoDelarte>
> Musician at ReverbNation.com <http://www.reverbnation.com/alonsodelarte>
>
> --
> Seqfan Mailing list - http://list.seqfan.eu/
>

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