Firoozbakht's conjecture

ATH · 2016-03-03, 19:35

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 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 = 10²⁷ 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).

science_man_88 · 2016-03-03, 20:15

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.

ATH · 2016-03-04, 10:46

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

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 $1)\: p_{n+1}-p_{n} < (log\: p_{n})^{2} - log\:p_{n}\: \: for\: \: n>4$ $2)\: \left (\frac{log(p_{n+1})}{log(p_{n})}\right )^{n} <\: e \:\: \Rightarrow \: p_{n+1} < p_{n}\^{e^{1/n}}$ $3)\: \left (\frac{p_{n+1}}{p_{n}}\right )^{n} <\: nlog(n)\: for\: n>5 \:\: \Rightarrow \: p_{n+1} < (nlog\: n)^{\frac{1}{n}} * p_{n}$ 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 = 10²⁷ 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

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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 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