Infinitely many?
 

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

Yes. Bredihin (Б. М. Бредихин) first proved this in 1963. See [url=https://oeis.org/A079545]A079545[/url].

Infinitely many?
 

[QUOTE=CRGreathouse;306141]Yes. Bredihin (Б. М. Бредихин) first proved this in 1963. See [url=https://oeis.org/A079545]A079545[/url].[/QUOTE]

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

[QUOTE=devarajkandadai;306417]Does this imply the infinitude of x^2 + 1?[/QUOTE]
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.

[QUOTE=devarajkandadai;306417]Does this imply the infinitude of x^2 + 1?[/QUOTE]

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.

Infinitely many?
 

[QUOTE=CRGreathouse;306421]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.[/QUOTE]
Would you care to have a look at my hueristic under the caption " draft proof" in PlanetMath?
tks

Link?

Infinitely many?
 

[QUOTE=CRGreathouse;306502]Link?[/QUOTE]
Will try & provide.
tks

Infinitely many?
 

[QUOTE=devarajkandadai;306592]Will try & provide.
tks[/QUOTE]

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