Timeline for Asymptotics on the number of ways to pair off $\{1, 2, \dots, 2n\}$ into primes
Current License: CC BY-SA 3.0
9 events
| when toggle format | what | by | license | comment | |
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| Jul 25, 2016 at 14:57 | comment | added | Bernardo Recamán Santos | If n = 12, then {(1,4), (2,5), (3,8), (6,7), (9,10), (11,12)} is the last time that such a pairing can be achieved in which pairs sum to distinct primes. | |
| Jul 20, 2016 at 15:03 | comment | added | Gerhard Paseman | One can note that for each odd number, there are something like $C\pi(2n)$ possible choices of even number to pair with it, where $C$ is some constant not much bigger than 1 (and probably less). Even for $C=2$ this gives an improvement ($(C\pi(n))^n$) on $n!$ for $n$ not too big. Gerhard "Can't Prove C Is Small" Paseman, 2016.07.20. | |
| Jul 20, 2016 at 10:40 | comment | added | Moritz Firsching | Perhaps an useful upper bound could be obtained by using the formulation as permanent together with the Bregman–Minc inequality. (Together with a useful estimate of the row sums from the prime number theorem). At least this should give something better than the trivial upper bound of $n!$. | |
| Jul 19, 2016 at 19:23 | history | edited | MT_ | CC BY-SA 3.0 |
added 4 characters in body
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| Jul 19, 2016 at 19:10 | answer | added | Gerhard Paseman | timeline score: 1 | |
| Jul 19, 2016 at 18:59 | comment | added | Gerhard Paseman | This observation is nice enough that I will partly spoil it: Use Bertrand to show that there is a prime p between 2n and 4n, and use that for one of the sums. The rest of the proof suggests a partial enumeration: say such a pairing is intervallic if the pairs that form the same sum q are an interval for any q. You might get a good asymptotic on the number of intervallic pairings. Gerhard "If Goldbach Were Easy" Paseman, 2016.07.19. | |
| Jul 19, 2016 at 18:30 | comment | added | Robert Israel | See OEIS sequence A000341. However, asymptotics are not given there. | |
| Jul 19, 2016 at 18:24 | review | First posts | |||
| Jul 19, 2016 at 18:25 | |||||
| Jul 19, 2016 at 18:21 | history | asked | MT_ | CC BY-SA 3.0 |