Do we know that A005574 is infinite?

[Allan Wechsler]

I've only done about ten minutes of research on this one. oeis.org/A005574 , at least from its graph, looks very copious. The number k is in the sequence if k + i is a Gaussian prime, and apparently we do not have a version of Dirichlet's theorem for arithmetic progressions of Gaussian primes. This is surprising to me.

Have I misinterpreted something? Is this sequence obviously infinite? If it's not, then I would think that the conjecture that it's infinite deserves a comment.

-- Allan

[Amiram Eldar]

This is Landau's 4th problem. Still unsolved after more than 100 years:

https://en.wikipedia.org/wiki/Landau%27s_problems

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[Allan Wechsler]

Thank you, Amiram! This fact is noted on oeis.org/A002496 (the actual primes of the form k^2 + 1), but not on oeis.org/A005574.

The four Landau problems are really a mathematical walk of shame. It's amazing how ignorant we are.

-- Allan

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[Ruud H.G. van Tol]

Hello Allan, I wonder what made you think of a "walk of shame" with
these four.

To me it looks easier to show that the 3x+1 problem is actually not a
problem, than to solve one of Landau's problems.
That certain problems are in classes like "obviously true, just not yet
proven" or maybe also like "even if somewhere false, just true enough in
practice", could well be part of the problem.

The Wiki-page also lists that there has been progress!

-- Greetings, Ruud

On 2026-02-03 23:04, Allan Wechsler wrote:
> Thank you, Amiram! This fact is noted on oeis.org/A002496

> <http://oeis.org/A002496> (the actual primes of the form k^2 + 1), but
> not on oeis.org/A005574 <http://oeis.org/A005574>.

>
> The four Landau problems are really a mathematical walk of shame. It's
> amazing how ignorant we are.
>
> -- Allan
>
> On Tue, Feb 3, 2026 at 4:56 PM Amiram Eldar <amiram...@gmail.com>
> wrote:
>
> This is Landau's 4th problem. Still unsolved after more than 100
> years:
> https://en.wikipedia.org/wiki/Landau%27s_problems
>
>
> On Tue, Feb 3, 2026 at 11:52 PM Allan Wechsler <acw...@gmail.com>
> wrote:
>
> I've only done about ten minutes of research on this one.

> oeis.org/A005574 <http://oeis.org/A005574> , at least from its

> graph, looks very copious. The number k is in the sequence if

> k + i is a Gaussian prime, and apparently we do /not /have a

> version of Dirichlet's theorem for arithmetic progressions of
> Gaussian primes. This is surprising to me.
>
> Have I misinterpreted something? Is this sequence obviously
> infinite? If it's not, then I would think that the conjecture
> that it's infinite deserves a comment.

> [...]
>

[David desJardins]

On Tue, Feb 3, 2026 at 9:16 PM 'Ruud H.G. van Tol' via SeqFan <seq...@googlegroups.com> wrote:

To me it looks easier to show that the 3x+1 problem is actually not a
problem, than to solve one of Landau's problems.

They seem like just different kinds of problems. It's not obvious which is harder. E.g., we have a proof that there is a prime between [n^3, (n+1)^3] for all sufficiently large n. So it's not obvious that [n^2, (n+1)^2] is intractable.

[Allan Wechsler]

Yes, Ruud, I probably could have been more careful in my wording. By "walk of shame" I just meant to dramatize how close we are in mathematics, at all times, to the complete unknown. I don't mean that we should be literally ashamed: these problems, after all, are as hard as they are. It's not like they would be any easier if we all led virtuous lives or anything.

I personally think that Landau's four problems are in some way more elegant than the Collatz conjecture. I think this is just my own esthetic sense. In particular I was surprised to learn that whatever techniques were used for the Dirichlet theorem on primes in arithmetic progressions could not be generalized to the setting of the Gaussian integers. Probably anybody who has ever actually been through the proof of the Dirichlet theorem can tell from this admission that I haven't.

I don't think I share your intuition that Collatz is easier than the Landau problems. I think I'm just agnostic on the subject.

I agree that the actual progress on these problems is fascinating and encouraging. The most recent dramatic example was Zhang's breakthrough on bounded gaps between primes (a weakening of the twin prime conjecture), followed by Tao and his gang driving the maximum possible gap size down and down -- not to 2, alas, but to something with 3 digits. I think this shows that real progress can be made by creatively weakening the various conjectures to see what we can prove. David desJardins gives another great example, of primes between consecutive cubes (almost solved) as a weakening of primes between consecutive squares (still almost completely stymied). One wonders if anybody is working on 2.5-th powers.

-- Allan

On Tue, Feb 3, 2026 at 9:16 PM 'Ruud H.G. van Tol' via SeqFan <seq...@googlegroups.com> wrote:

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[Gareth McCaughan]

On 04/02/2026 02:55, Allan Wechsler wrote:

I personally think that Landau's four problems are in some way more elegant than the Collatz conjecture. I think this is just my own esthetic sense. In particular I was surprised to learn that whatever techniques were used for the Dirichlet theorem on primes in arithmetic progressions could not be generalized to the setting of the Gaussian integers. Probably anybody who has ever actually been through the proof of the Dirichlet theorem can tell from this admission that I haven't.

The sort of generalization of Dirichlet to Gaussian integers that I'd naïvely expect looks different from the one you naïvely expect :-). I'd expect to see something more like this: suppose you have a Gaussian integer M; there are finitely many residue classes of "Gaussian integers mod M"; there are "about equally many" Gaussian primes in all of those. Note that the sets we're expecting to find Gaussian integers in here are (in the complex plane) 2D lattices rather than 1D ones.

I have no idea whether there is any such generalization; a bit of casual googling didn't turn anything up. One possible obstacle to there being one is that if there is then you'd expect it to apply to other quadratic fields besides Q(i), and for some of those the ring of integers is no longer a principal ideal domain and you're supposed to talk about prime _ideals_ rather than prime _numbers_, and I don't know even how to state an appropriate generalization, never mind how one might try to prove it.

There's a thing called the Chebotarev density theorem that allegedly has Dirichlet's theorem as a special case (albeit in a somewhat subtle way that I don't really understand). It doesn't _look_ to me as if it obviously implies anything like the above handwavy thing, but again I don't really understand it. Do we have any actual number theorists here?

I don't think I share your intuition that Collatz is easier than the Landau problems. I think I'm just agnostic on the subject.

I agree that the actual progress on these problems is fascinating and encouraging. The most recent dramatic example was Zhang's breakthrough on bounded gaps between primes (a weakening of the twin prime conjecture), followed by Tao and his gang driving the maximum possible gap size down and down -- not to 2, alas, but to something with 3 digits. I think this shows that real progress can be made by creatively weakening the various conjectures to see what we can prove. David desJardins gives another great example, of primes between consecutive cubes (almost solved) as a weakening of primes between consecutive squares (still almost completely stymied). One wonders if anybody is working on 2.5-th powers.

For what little it's worth, I agree with all that.

--
g