Timeline for Are there at least one of $a$, $b$, $c$, $a+b$, $b+c$, $c+a$ have h(P) $\le 3$? (was checked up to $10^{18}$)
Current License: CC BY-SA 4.0
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| S Jul 12, 2020 at 7:02 | history | bounty ended | CommunityBot | ||
| S Jul 12, 2020 at 7:02 | history | notice removed | CommunityBot | ||
| Jul 4, 2020 at 6:27 | comment | added | Seva | I wonder whether one can prove at least that there exists an absolute constant $k$ such that the set of all $k$-ful numbers does not have any Schur triples. | |
| S Jul 4, 2020 at 5:47 | history | bounty started | Đào Thanh Oai | ||
| S Jul 4, 2020 at 5:47 | history | notice added | Đào Thanh Oai | Authoritative reference needed | |
| Jun 15, 2020 at 7:27 | history | edited | CommunityBot |
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| S Jun 2, 2020 at 18:03 | history | bounty ended | CommunityBot | ||
| S Jun 2, 2020 at 18:03 | history | notice removed | CommunityBot | ||
| May 26, 2020 at 7:54 | comment | added | Yaakov Baruch | @SylvainJULIEN: the corresponding density for squarefull numbers is $\zeta(3/2)/\zeta(3) N^{1/2}\approx 2.173254 N^{1/2}$. I would imagine that for cubefull numbers the constant similarly relates to values of $\zeta$. | |
| May 25, 2020 at 16:47 | comment | added | Sylvain JULIEN | @Yaakov Baruch: could there be a heuristics related to bifurcation theory suggesting that your $4.67$ actually is the first Feigenbaum constant? | |
| S May 25, 2020 at 16:23 | history | bounty started | Đào Thanh Oai | ||
| S May 25, 2020 at 16:23 | history | notice added | Đào Thanh Oai | Authoritative reference needed | |
| Oct 25, 2019 at 8:17 | comment | added | Đào Thanh Oai | The conjecture is a generalization of the Fermat last theorem, Beal conjecture, Fermat-Catalan conjecture and I think maybe equivalent to ABC conjecture | |
| Oct 24, 2019 at 11:06 | comment | added | Yaakov Baruch | @Đào Thanh Oai. There is nothing in my calculations that warrants publication. I also think the conjecture itself is one of many many that are very likely to be true (probabilistically, once checked for small values) but are in the same general order of difficulty to prove as an effective $abc$-conjecture. Most such conjectures seem to be stated as dead-end curiosities, not leading to some deeper or wider insight. | |
| Oct 18, 2019 at 5:30 | history | edited | Martin Sleziak | CC BY-SA 4.0 |
MathJax: \gcd
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| Oct 18, 2019 at 2:59 | history | edited | Đào Thanh Oai | CC BY-SA 4.0 |
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| Oct 7, 2019 at 2:14 | comment | added | Đào Thanh Oai | @YaakovBaruch From your work, can you publish the good conjecture on ArXvi or any journal? | |
| Oct 4, 2019 at 3:00 | comment | added | Greg Martin | Yes, I believe the counting function for numbers with $h(n)\ge k$ is $\sim c_k x^{1/k}$. This is trivially true for $k=1$ and well known for $k=2$, and I think the latter proof generalizes to any $k$. | |
| Oct 4, 2019 at 1:49 | history | edited | Đào Thanh Oai | CC BY-SA 4.0 |
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| Oct 4, 2019 at 1:29 | history | edited | Đào Thanh Oai | CC BY-SA 4.0 |
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| Oct 4, 2019 at 1:24 | history | edited | Đào Thanh Oai | CC BY-SA 4.0 |
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| Oct 3, 2019 at 19:54 | comment | added | Yaakov Baruch | Empirically, with $N=10^n, n=1,2,\dots 21$, I suspect something like $\approx 4.67 N^{1/3}$. But some of the references at OEIS A036966 may be addressing this... | |
| Oct 3, 2019 at 18:38 | comment | added | Yaakov Baruch | What is the density of the set with $P\le N, h(P)\ge 3$ asymptotically? (I would expect an answer of the form $cN^a$ for specific positive reals $a<1, c$.) | |
| Oct 3, 2019 at 14:58 | history | asked | Đào Thanh Oai | CC BY-SA 4.0 |