mersenneforum.org > Math Firoozbakht's conjecture
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 2016-03-03, 19:35 #1 ATH Einyen     Dec 2003 Denmark 1100000000102 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 $1)\: p_{n+1}-p_{n} < (log\: p_{n})^{2} - log\:p_{n}\: \: for\: \: n>4$ $2)\: \left (\frac{log(p_{n+1})}{log(p_{n})}\right )^{n} <\: e \:\: \Rightarrow \: p_{n+1} < p_{n}\^{e^{1/n}}$ $3)\: \left (\frac{p_{n+1}}{p_{n}}\right )^{n} <\: n*log(n)\: for\: n>5 \:\: \Rightarrow \: p_{n+1} < (n*log\: n)^{\frac{1}{n}} * p_{n}$ I noticed that except for very small values of pn (2) is actually stronger than (1) contrary to what is listed, and (3) is even stronger than (2). I calculated the max pn+1 allowed in the 3 conjectures for values in the p(10^x) and p(2^x) tables using mpfr. For example: pn = 1027 and n=16352460426841680446427399: (1) pn+1 - pn < 3802.9139250279 (2) pn+1 - pn < 3801.8619759990 (3) pn+1 - pn < 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 pn = 1447 (n=229), and (2) is weaker than (1) for many values up to and including pn = 11801 (n=1414). (3) is always stronger than (2) for pn > 5. After this (1) is weaker than (2) which is weaker than (3) up to at least pn = 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 science_man_88     "Forget I exist" Jul 2009 Dumbassville 26×131 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 ATH Einyen     Dec 2003 Denmark 2·29·53 Posts If we insert pn = 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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