Timeline for Are there infinitely many karmic numbers, i.e numbers whose primality radii equal one or a prime power?
Current License: CC BY-SA 4.0
15 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Nov 25, 2019 at 21:20 | answer | added | Sylvain JULIEN | timeline score: -1 | |
| Nov 25, 2019 at 11:32 | comment | added | Sylvain JULIEN | Thank you very much Daniel. I found an article about the prime factors in an arithmetic progression that should lead to a proof that there are finitely many karmic numbers divisible by 6. I'll try to post it as a partial answer. | |
| Nov 25, 2019 at 7:21 | comment | added | Daniel McLaury | I can confirm that the non-primes up to 20 together with 24, 30, 34, 36, 42, and 60 are the only examples up to $10^8$. | |
| Nov 25, 2019 at 7:08 | comment | added | Sylvain JULIEN | I may have an idea that might lead to a proof later: let $(r(i))_{1\leq i\leq n}$ be a strictly increasing arithmetic progression of positive integers. One may try to show that in any dense enough subsequence thereof, for example whose number of terms is $\Omega(n^{1-\delta})$ for some $0<\delta<1$, the arithmetic function $\omega$ takes at least 3 different values provided $n$ is large enough. | |
| Nov 25, 2019 at 6:48 | comment | added | Steven Landsburg | To be clear about the logic of the comment from @ThomasSauvaget : The OP asks if there are infinitely many $q$ such that for all $r$, $A(q,r)$ implies either $B(r)$ or $C(r)$. The comment claims this is equivalent to asking if there are infinitely many $q$ such that there exists an $r$ satisfying both $A(q,r)$ and either $B(r)$ or $C(r)$. It isn't. | |
| Nov 25, 2019 at 5:47 | comment | added | Sylvain JULIEN | Thank you Robert. The numbers $1$, $2$, $3$, $5$ and $7$ have to removed from the list though, as $1$ has no primality radius, while the four other integers are prime and thus admit $0$ as a primality radius, which is neither equal to $1$ nor to a prime power. | |
| Nov 25, 2019 at 0:48 | comment | added | Robert Israel | If I haven't made a mistake, the only karmic numbers $< 10^5$ are $1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 16, 18, 20, 24, 30, 34, 36, 42, 60$. The sequence does not appear to be in the OEIS. I would guess that these are the only karmic numbers, but a proof would be difficult. | |
| Nov 24, 2019 at 20:17 | comment | added | Sylvain JULIEN | I asked several questions related to the notion of primality radius on the assumption of Goldbach's conjecture, which is equivalent to the existence of a primality radius for all integer greater than $1$. A prime has trivially a primality radius equal to 0, so I discard this possibility as uninteresting. | |
| Nov 24, 2019 at 20:07 | comment | added | Daniel McLaury | I didn't understand the reason for the condition that $n$ is composite, or that it is "sufficiently large." Also, what have you done so far? | |
| Nov 24, 2019 at 19:50 | comment | added | Sylvain JULIEN | Good point, Ilya. It must be a matter of quantifiers, as it seems Thomas used "exists" while I meant "for all". | |
| Nov 24, 2019 at 19:16 | comment | added | Ilya Bogdanov | @ThomasSauvaget: I do not think your formulation is equivalent to OP's. For your pair, the number (p_1+p_2)/2$ may have another radii which fail to be of the required form. | |
| Nov 24, 2019 at 18:37 | comment | added | Thomas Sauvaget | I disagree with your assessment. Adding unecessary terminology or hypotheses is very un-mathematical. The formulation isn't dry, it just requires to understand the object at hand by looking at examples and related theorems to patiently build a mental picture. | |
| Nov 24, 2019 at 18:20 | comment | added | Sylvain JULIEN | Introducing such a terminology may help lead to further investigations in the future and also allows the formulation to be less dry. Math is not only a matter of reasoning but also of sensitivity. But indeed your reformulation is correct. | |
| Nov 24, 2019 at 18:13 | comment | added | Thomas Sauvaget | You are asking if there are infinitely many pairs of primes $(p_1,p_2)$ such that either $p_2=p_1+2$ or $p_2=p_1+2q^k$ for some prime $q$ and integer $k$. There is no need to introduce either a "primality radius" nor a "karmic number". | |
| Nov 24, 2019 at 17:35 | history | asked | Sylvain JULIEN | CC BY-SA 4.0 |