Timeline for Can anything deep be said uniformly about conjectures like Goldbach's?
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| Mar 5, 2019 at 16:10 | comment | added | user21820 | I think you've misunderstood me from the very beginning. I'm not at all interested in formulating novel interesting conjectures. Neither do I care whether my random conjectures have been investigated before. What I asked clearly was whether there is any reason to believe that there are deep uniform reasons for such conjectures that have no small counter-examples. That's all. | |
| Mar 5, 2019 at 16:00 | comment | added | Zhi-Wei Sun | I don't agree your viewpoint. What I mean is that it is not easy to formulate a really novel and interesting conjecture about primes that has not been investigated before. | |
| Mar 5, 2019 at 13:54 | comment | added | user21820 | Thanks. Anyway I have no idea why you keep saying it's not new. I don't care whether it's new; it was just an example to show how easy it is to come up with conjectures of that kind where minor tweaks have relatively large counter-examples, and as I said in my first comment my question has always been about the small cases, not about the asymptotic behaviour. (And I upvoted your answer so I can't upvote it again...) | |
| Mar 5, 2019 at 12:36 | history | edited | Zhi-Wei Sun | CC BY-SA 4.0 |
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| Mar 5, 2019 at 12:33 | comment | added | Zhi-Wei Sun | See oeis.org/A065377 for the list of primes not of the form $p+k^2$ with $k>0$. Robert G. Wilson v wrote on Nov 05 2001 that "Probably finite and 7549 is the last entry." So I do consider your PSQ not new at all | |
| Mar 5, 2019 at 12:30 | history | edited | Zhi-Wei Sun | CC BY-SA 4.0 |
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| Mar 5, 2019 at 11:59 | comment | added | user21820 | Yes, but I was just saying that I cannot use the OEIS-cited search to claim that there are no counter-examples to my conjecture up to 3 billion. Asymptotically, I'd expect the number of decompositions to tend to infinity, as you say. | |
| Mar 5, 2019 at 11:38 | comment | added | Zhi-Wei Sun | Actually, Hardy and Littlewood had a conjecture on the main term of the number $r(n)$ of ways to write a large non-square integer $n$ as the sum of a prime and a sqaure. According to their conjecture, $r(n)\to\infty$. This, of course, implies that large prime can be written as the sum of a prime and a positive square. | |
| Mar 5, 2019 at 7:05 | comment | added | user21820 | Oh thanks yea I didn't see that. My results match theirs for non-primes, but I require a positive square so unfortunately their search doesn't cover primes. | |
| Mar 5, 2019 at 6:57 | comment | added | Zhi-Wei Sun | It seems that you have not yet visited oeis.org/A020495 which lists all the known 21 terms which are neither squares nor primes plus squares. In the Extension part, it wrotes that "Almost certainly finite; no other terms below 25000000. Search extended to 3000000000 by James Van Buskirk without finding any more terms". - John Robertson (Jpr2718(AT)aol.com) (2009) | |
| Mar 5, 2019 at 6:41 | comment | added | user21820 | Thanks, but as with Timothy's cited conjecture, this concerns asymptotic behaviour, whereas I'm asking about the small cases, i.e. below the bound where the asymptotic behaviour takes over. | |
| Mar 5, 2019 at 3:22 | history | edited | Zhi-Wei Sun | CC BY-SA 4.0 |
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| Mar 5, 2019 at 3:05 | history | edited | Zhi-Wei Sun | CC BY-SA 4.0 |
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| Mar 5, 2019 at 2:59 | history | answered | Zhi-Wei Sun | CC BY-SA 4.0 |