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 2008-03-31, 04:50 #1 MooooMoo Apprentice Crank     Mar 2006 2·227 Posts Connection between rank and digits on top5000 list I was looking through the top 5000 primes list yesterday, and I found an interesting connection between a prime's rank and the number of digits it has. Let's call it "Moo's law". Moo's law states that the rank of the prime to the power of 0.4 multiplied by the number of digits of the prime will always give rougly the same number. Let's see how it works: For the top 10 primes, Moo's law doesn't work too well. The top 5 primes are simply too big. Code: Rank Prime Rank^0.4 Digits Rank^0.4 * Digits 1 2^32582657-1 1 9,808,358 9,808,358 2 2^30402457-1 1.3195 9,152,052 12,076,205 3 2^25964951-1 1.5518 7,816,230 12,129,582 4 2^24036583-1 1.7411 7,235,733 12,598,143 5 2^20996011-1 1.9037 6,320,430 12,031,011 6 2^13466917-1 2.0477 4,053,946 8,301,154 7 19249*2^13.. 2.1779 3,918,990 8,535,194 8 27653*2^91.. 2.2974 2,759,677 6,340,073 9 28433*2^78.. 2.4082 2,357,207 5,676,684 10 33661*2^70.. 2.5119 2,116,617 5,316,702 However, Moo's law works really well for the non-megabit primes (ranks 15-5000). Nearly all of them give a rank^0.4 * digits value that is between 2.5 million and 3 million: Code: Rank Rank^0.4 Digits Rank^0.4 * Digits 15 2.9542 999,997 2,954,168 18 3.1777 909,526 2,890,175 25 3.6239 794,556 2,879,390 40 4.3734 644,179 2,817,283 50 4.7818 560,439 2,679,886 75 5.6237 464,957 2,614,793 100 6.3096 407,850 2,573,360 200 8.3256 301,252 2,508,083 500 12.0112 208,937 2,509,593 750 14.1262 192,241 2,715,630 1000 15.8489 169,014 2,678,691 2000 20.9128 126,660 2,648,814 3000 24.5951 112,619 2,769,875 4000 27.5946 108,344 2,989,709 4500 28.9258 101,539 2,937,094 I don't know whether anyone else has noticed this before, but it sure is an interesting find.
2008-03-31, 05:00   #2
retina
Undefined

"The unspeakable one"
Jun 2006
My evil lair

11010000101102 Posts

Quote:
 Originally Posted by MooooMoo ... always give rougly the same number.
Any chance of a more mathematical description than "rougly"? Actually I don't even know what "rougly" means, maybe it is a mathematical term?

But I am sure you are on to something really really important there, can it be used to predict the upcoming 45th? Or maybe help towards an end global warming (if it is really warming that is)?

2008-03-31, 05:25   #3
MooooMoo
Apprentice Crank

Mar 2006

2×227 Posts

Quote:
 Originally Posted by retina Any chance of a more mathematical description than "rougly"? Actually I don't even know what "rougly" means, maybe it is a mathematical term?
I meant "roughly", or "approximately". It was a typo; I should have used my computer's spell check.

2008-03-31, 05:45   #4
retina
Undefined

"The unspeakable one"
Jun 2006
My evil lair

2·32·7·53 Posts

Quote:
 Originally Posted by MooooMoo I meant "roughly", or "approximately". It was a typo; I should have used my computer's spell check.
I use Firefox with the language add-on. It nicely highlights anything it doesn't recognise with a red underline.

Hehe, Firefox doesn't like the word "Firefox". Silly thing.

2008-03-31, 16:31   #5
victor

Oct 2005
Fribourg, Switzerlan

3748 Posts

Quote:
 Originally Posted by retina I use Firefox with the language add-on. It nicely highlights anything it doesn't recognise with a red underline. Hehe, Firefox doesn't like the word "Firefox". Silly thing.
Let's call it "Retina's law".

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