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Old 2012-07-27, 03:27   #1
devarajkandadai
 
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Default Infinitely many?

Are there infinitely many primes of the form x^2 + y^2 + 1 ?
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Old 2012-07-27, 03:56   #2
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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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Old 2012-07-30, 04:29   #3
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Originally Posted by CRGreathouse View Post
Yes. Bredihin (Б. М. Бредихин) first proved this in 1963. See A079545.
Does this imply the infinitude of x^2 + 1?
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Old 2012-07-30, 05:10   #4
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Quote:
Originally Posted by devarajkandadai View Post
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.

Last fiddled with by LaurV on 2012-07-30 at 05:11
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Old 2012-07-30, 05:17   #5
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Quote:
Originally Posted by devarajkandadai View Post
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.
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Old 2012-07-31, 04:07   #6
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Originally Posted by CRGreathouse View Post
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
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Old 2012-07-31, 06:38   #7
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Link?
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Old 2012-08-01, 04:23   #8
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Default Infinitely many?

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Originally Posted by CRGreathouse View Post
Link?
Will try & provide.
tks
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Old 2012-08-01, 04:31   #9
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Default Infinitely many?

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Originally Posted by devarajkandadai View Post
Will try & provide.
tks
PlanetMath.org - Forums- Research/postgraduate/Draft proof

Trust above is sufficient.
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