Infinitely many?

devarajkandadai · 2012-07-27, 03:27

Are there infinitely many primes of the form x^2 + y^2 + 1 ?

CRGreathouse · 2012-07-27, 03:56

Yes. Bredihin (Б. М. Бредихин) first proved this in 1963. See A079545.

devarajkandadai · 2012-07-30, 04:29

Quote:

Originally Posted by CRGreathouse

Yes. Bredihin (Б. М. Бредихин) first proved this in 1963. See A079545.

Does this imply the infinitude of x^2 + 1?

LaurV · 2012-07-30, 05:10

Quote:

Originally Posted by devarajkandadai

Does this imply the infinitude of x^2 + 1?

Not really. It may be that after a while, all primes of the form x^2+y^2+1 have x and y bigger than 0. I know the problem you mention is still open.

CRGreathouse · 2012-07-30, 05:17

Quote:

Originally Posted by devarajkandadai

Does this imply the infinitude of x^2 + 1?

No, this is a much harder problem. We believe that there are infinitely many of that form, and Hardy & Littlewood have a conjecture giving the precise density of such primes, but there's no proof yet. I don't think it will come soon, either. Green & Tao have made great progress, it seems, on the linear case (Dickson's conjecture) but the higher-degree case remains open AFAICT.

devarajkandadai · 2012-07-31, 04:07

Quote:

Originally Posted by CRGreathouse

No, this is a much harder problem. We believe that there are infinitely many of that form, and Hardy & Littlewood have a conjecture giving the precise density of such primes, but there's no proof yet. I don't think it will come soon, either. Green & Tao have made great progress, it seems, on the linear case (Dickson's conjecture) but the higher-degree case remains open AFAICT.

Would you care to have a look at my hueristic under the caption " draft proof" in PlanetMath?
tks

CRGreathouse · 2012-07-31, 06:38

Link?

devarajkandadai · 2012-08-01, 04:23

Quote:

Originally Posted by CRGreathouse

Link?

Will try & provide.
tks

devarajkandadai · 2012-08-01, 04:31

Quote:

Originally Posted by devarajkandadai

Will try & provide.
tks

PlanetMath.org - Forums- Research/postgraduate/Draft proof

Trust above is sufficient.

2012-07-27, 03:27	#1
devarajkandadai May 2004 2²×79 Posts	Infinitely many? Are there infinitely many primes of the form x^2 + y^2 + 1 ?

2012-07-27, 03:56	#2
CRGreathouse Aug 2006 2²×1,493 Posts	Yes. Bredihin (Б. М. Бредихин) first proved this in 1963. See A079545. Last fiddled with by CRGreathouse on 2012-07-27 at 04:09

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2012-07-31, 06:38	#7
CRGreathouse Aug 2006 2²×1,493 Posts	Link?