2016-03-03, 19:35 | #1 |
Einyen
Dec 2003
Denmark
110000000010_{2} Posts |
Firoozbakht's conjecture
I found this conjecture along with 2 related conjectures concerning prime gaps:
https://en.wikipedia.org/wiki/Firooz...27s_conjecture https://oeis.org/A182514 I noticed that except for very small values of p_{n} (2) is actually stronger than (1) contrary to what is listed, and (3) is even stronger than (2). I calculated the max p_{n+1} allowed in the 3 conjectures for values in the p(10^x) and p(2^x) tables using mpfr. For example: p_{n} = 10^{27} and n=16352460426841680446427399: (1) p_{n+1} - p_{n} < 3802.9139250279 (2) p_{n+1} - p_{n} < 3801.8619759990 (3) p_{n+1} - p_{n} < 3798.6843745827 I tested with mpfr precision 256 bit and 512 bit and the results only changed somewhere after the 50th decimal, so I used 256 bit precision. When I tested small values I found (3) is weaker than (1) for many values up to and including p_{n} = 1447 (n=229), and (2) is weaker than (1) for many values up to and including p_{n} = 11801 (n=1414). (3) is always stronger than (2) for p_{n} > 5. After this (1) is weaker than (2) which is weaker than (3) up to at least p_{n} = 10^8 and for all known p(2^x) and p(10^x) values up to p(2^86) and p(10^27). Last fiddled with by ATH on 2016-03-03 at 19:37 |
2016-03-03, 20:15 | #2 |
"Forget I exist"
Jul 2009
Dumbassville
2^{6}×131 Posts |
http://mersenneforum.org/forumdisplay.php?f=131 < prime gap searches forum
and specifically: http://mersenneforum.org/showthread.php?t=21045 you may be able to relate and see how good it is. |
2016-03-04, 10:46 | #3 |
Einyen
Dec 2003
Denmark
2·29·53 Posts |
If we insert p_{n} = n*log n in (1), (2) and (3) and if Wolfram Alpha is correct, this shows that (1) grows more slowly and both (2) and (3):
http://www.wolframalpha.com/input/?i...+n-%3Einfinity http://www.wolframalpha.com/input/?i...+n-%3Einfinity So (1) is actually stronger than (2) and (3) for n->infinity ? (2) and (3) grows at about the same rate: http://www.wolframalpha.com/input/?i...+n-%3Einfinity |
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