Timeline for Is every odd positive integer of the form $P_{n+m}-P_n-P_m$?
Current License: CC BY-SA 4.0
63 events
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| Nov 1, 2023 at 1:25 | comment | added | Đào Thanh Oai | Yes, you are right. I don't know what Conjecture 1 has to do with Goldbach's conjecture, may you help me? @GerryMyerson | |
| Oct 31, 2023 at 21:20 | comment | added | Gerry Myerson | If $y$ is a sum of two primes, and a difference of two odd numbers, then $y$ is even, not odd. | |
| Oct 31, 2023 at 16:05 | comment | added | Đào Thanh Oai | @GerryMyerson Today, I see that maybe conjecture 1 equivalent to Goldbach's conjecture en.wikipedia.org/wiki/Goldbach%27s_conjecture or Corollary of the Goldbach's conjecture, because $P_n+P_m=P_{n+m}-x=y$. Where $x$ is an odd positive integer less than $P_{n+m}$ then $y$ is an odd positive integer less $P_{n+m}$. But maybe we must prove that $P_n+P_m \le P_{m+n}$. Is my remark right? | |
| Jun 15, 2020 at 7:27 | history | edited | CommunityBot |
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| Jul 9, 2018 at 5:02 | vote | accept | Đào Thanh Oai | ||
| Jul 9, 2018 at 5:02 | vote | accept | Đào Thanh Oai | ||
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| Jul 8, 2018 at 22:44 | vote | accept | Đào Thanh Oai | ||
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| Jul 8, 2018 at 22:44 | vote | accept | Đào Thanh Oai | ||
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| Jul 8, 2018 at 22:44 | vote | accept | Đào Thanh Oai | ||
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| Jul 8, 2018 at 22:43 | vote | accept | Đào Thanh Oai | ||
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| Jul 8, 2018 at 22:43 | vote | accept | Đào Thanh Oai | ||
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| Jul 8, 2018 at 15:43 | vote | accept | Đào Thanh Oai | ||
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| Jul 8, 2018 at 15:41 | vote | accept | Đào Thanh Oai | ||
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| Jul 8, 2018 at 15:41 | vote | accept | Đào Thanh Oai | ||
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| S Jul 8, 2018 at 15:40 | history | bounty ended | Đào Thanh Oai | ||
| S Jul 8, 2018 at 15:40 | history | notice removed | Đào Thanh Oai | ||
| Jul 6, 2018 at 5:29 | vote | accept | Đào Thanh Oai | ||
| Jul 8, 2018 at 15:41 | |||||
| Jul 5, 2018 at 18:43 | answer | added | Aaron Meyerowitz | timeline score: 2 | |
| Jul 5, 2018 at 0:54 | comment | added | Đào Thanh Oai | @GHfromMO I hope that. Thank You very much | |
| Jul 4, 2018 at 11:34 | comment | added | GH from MO | Just some remarks. I believe that Conjectures 1 and 3 are out of reach at present by being "binary additive problems in the primes". Conjecture 2 is straightforward by being a variant of the ternary Goldbach problem. See also the following related MO entries: mathoverflow.net/questions/202979/… and mathoverflow.net/questions/253324/… | |
| Jul 4, 2018 at 1:41 | history | edited | Đào Thanh Oai | CC BY-SA 4.0 |
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| Jul 3, 2018 at 16:57 | comment | added | Pierre-Yves Gaillard |
@ĐàoThanhOai - The usual terminology is to call elements of $\mathbb Z$ integers, and to say that an integer $n\in\mathbb Z$ is nonnegative if $n\ge0$ and positive if $n>0$. Also, if you want a user to be notified, you can use the @ symbol.
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| Jul 3, 2018 at 16:45 | comment | added | Đào Thanh Oai | I mean positive integer, can You help me correct? | |
| Jul 3, 2018 at 16:41 | comment | added | Pierre-Yves Gaillard | By "integer", do you mean "positive integer", or do you really mean integer? | |
| Jul 3, 2018 at 16:15 | history | edited | Đào Thanh Oai | CC BY-SA 4.0 |
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| Jul 3, 2018 at 15:48 | answer | added | LeechLattice | timeline score: 16 | |
| Jul 3, 2018 at 15:26 | history | edited | Đào Thanh Oai | CC BY-SA 4.0 |
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| Jul 3, 2018 at 15:07 | history | edited | Đào Thanh Oai | CC BY-SA 4.0 |
Conjecture simple but stronger
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| Jul 3, 2018 at 15:02 | comment | added | Đào Thanh Oai | I have just computed the conjecture 3 is true with $x=2, 4, \cdots, 10^6$ and $3873$ numbers from $[982197492$, $982197494$, $\cdots$, $982226054]$=$[9,82197492.10^8$, $\cdots$, $982226054.10^8]$ | |
| Jul 3, 2018 at 15:01 | history | edited | Đào Thanh Oai | CC BY-SA 4.0 |
Conjecture simple but stronger
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| Jul 3, 2018 at 13:12 | history | edited | Đào Thanh Oai | CC BY-SA 4.0 |
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| Jul 3, 2018 at 12:54 | history | edited | Đào Thanh Oai | CC BY-SA 4.0 |
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| Jul 3, 2018 at 12:48 | history | edited | Đào Thanh Oai | CC BY-SA 4.0 |
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| Jul 3, 2018 at 8:31 | history | edited | Đào Thanh Oai | CC BY-SA 4.0 |
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| Jul 3, 2018 at 8:00 | history | edited | Đào Thanh Oai | CC BY-SA 4.0 |
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| Jul 3, 2018 at 7:42 | comment | added | Đào Thanh Oai | @LevBorisov Please see the question above. I wish You are co-author of the general case, do You agree? | |
| Jul 3, 2018 at 7:40 | history | edited | Đào Thanh Oai | CC BY-SA 4.0 |
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| Jul 3, 2018 at 0:46 | comment | added | Đào Thanh Oai | I agree that may be, every odd integer $x$ of the form $P_{m+n+1}-P_m-P_n$. Thank You for your help @LevBorisov | |
| Jul 2, 2018 at 19:50 | comment | added | Lev Borisov | It is a nice observation. I doubt that there is any way to find a proof, since the number of a prime is a rather fickle invariant. I suspect that if you ask about $P_{m+n+1}-P_m-P_n$ the results might be similar. It might be more reasonable to ask whether one can prove that supremum over $x$ of minimum $|a+b-c|$ for all $x=P_c-P_a-P_b$ is bounded. This is a notably weaker statement, but it seems to me to be more hopeful. | |
| Jul 2, 2018 at 17:59 | comment | added | Đào Thanh Oai | Now I checked this is true with $x=1,3,⋯,54373881$; $(54373881 = 5.4373881*10^7)$ @Zhi-WeiSun | |
| Jul 2, 2018 at 12:27 | comment | added | LSpice | @MattF., that was my edit in the title, and I forgot the old maxim that 'any' should never be used as a quantifier, because it can easily be read both as 'every' and as 'some'. I agree that the current title is better. | |
| Jul 2, 2018 at 8:06 | comment | added | Đào Thanh Oai | @Zhi-WeiSun Can I publish this conjecture on your journal? | |
| Jul 2, 2018 at 5:10 | comment | added | Đào Thanh Oai | @Zhi-WeiSun Thank You very much, but with m=2 it is very difficulte checked with $x$ large. I have a algorithm to fast checked this result. If You can help me I will comunication with You. | |
| Jul 2, 2018 at 5:06 | comment | added | Zhi-Wei Sun | I agree Wojowu's comment: Conjectures on prime constellations should imply this is possible even with $m=2$. | |
| Jul 2, 2018 at 3:26 | comment | added | Đào Thanh Oai | Yes, thank You very much. I edited again. I have just checked this true with x=1, 3,....12342955 @MattF. | |
| Jul 2, 2018 at 1:27 | history | edited | Đào Thanh Oai | CC BY-SA 4.0 |
edited title
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| Jul 2, 2018 at 1:24 | comment | added | user44143 | "Is every odd integer..." would be a version of the title more consistent with the question. | |
| Jul 1, 2018 at 16:25 | comment | added | Đào Thanh Oai | Yes, there are many numbers $m, n$ so that $P_{m+n}-Pm-P_n=10^7-1$ | |
| Jul 1, 2018 at 16:20 | comment | added | LSpice | Indeed it seems that there is no need to link to a text file to give the simply stated answer to @GerhardPaseman's question. As your file indicates, $m = 6787300$ and $n = 6787562$ (not much larger than $10^7$, for reasonable values of "not much larger"), so $P_m = 118988791$, $P_n = 118993759$, and $P_{m + n} = 247982549$. | |
| S Jul 1, 2018 at 15:08 | history | bounty started | Đào Thanh Oai | ||
| S Jul 1, 2018 at 15:08 | history | notice added | Đào Thanh Oai | Authoritative reference needed | |
| Jun 30, 2018 at 23:10 | comment | added | Gerhard Paseman | Thank you. I prefer to just have a single pair of values. In the meantime, I have convinced myself that m and n exist and should not be much larger than 10^7. Gerhard "Thanks Again For Your Link" Paseman, 2018.06.30. | |
| Jun 29, 2018 at 1:27 | comment | added | Đào Thanh Oai | @GerhardPaseman When You ask, it is 1 AM in my country, I was going to sleep. So now the answer. Click link as follows to view. In the file, I define that. $Amn=P_{m+n}, Am=P_{m}, An=P_n, C=P_{m+n}-P_m-P_n$ the answer in this text file | |
| Jun 28, 2018 at 16:36 | comment | added | Wojowu | Conjectures on prime constellations should imply this is possible even with $m=2$. | |
| Jun 28, 2018 at 15:31 | comment | added | Đào Thanh Oai | I am sorry, I didn't print $m, n$ so please waiting me. @GerhardPaseman | |
| Jun 28, 2018 at 15:13 | comment | added | Gerhard Paseman | So what values of n and m gave rise to the x that is 10^7-1? Gerhard "Suspect A Typo In Exponent" Paseman, 2018.06.28. | |
| Jun 28, 2018 at 13:35 | comment | added | Đào Thanh Oai | I’m sorry, I think the result can checked by Computer | |
| Jun 28, 2018 at 13:14 | history | edited | GH from MO |
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| Jun 28, 2018 at 13:06 | comment | added | Gerry Myerson | Why the "computer science" tag? Why the "prime ideals" tag? You know, there's no law that says you must use five tags – you're allowed to use fewer. | |
| Jun 28, 2018 at 11:55 | history | edited | LSpice | CC BY-SA 4.0 |
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| Jun 28, 2018 at 11:21 | history | edited | Đào Thanh Oai | CC BY-SA 4.0 |
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| Jun 28, 2018 at 11:13 | comment | added | Stanley Yao Xiao | You might as well assume that $m \geq 2$, since otherwise $x$ is even. | |
| Jun 28, 2018 at 10:58 | history | asked | Đào Thanh Oai | CC BY-SA 4.0 |