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Old 2016-03-03, 19:35   #1
ATH
Einyen
 
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Default 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
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Old 2016-03-03, 20:15   #2
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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.
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Old 2016-03-04, 10:46   #3
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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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