Connection between rank and digits on top5000 list

[MooooMoo]

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.

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:

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.

[retina]

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)?

[MooooMoo]

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.

[retina]

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.

[victor]

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".