New Pattern Found in Prime Numbers (Generalized Benford's Law)

[phantom power]

Benford’s law (BL), named after physicist Frank Benford in 1938, describes the distribution of the leading digits of the numbers in a wide variety of data sets and mathematical sequences. Somewhat unexpectedly, the leading digits aren’t randomly or uniformly distributed, but instead their distribution is logarithmic. That is, 1 as a first digit appears about 30% of the time, and the following digits appear with lower and lower frequency, with 9 appearing the least often. Benford’s law has been shown to describe disparate data sets, from physical constants to the length of the world’s rivers.

Since the late ‘70s, researchers have known that prime numbers themselves, when taken in very large data sets, are not distributed according to Benford’s law. Instead, the first digit distribution of primes seems to be approximately uniform. However, as Luque and Lacasa point out, smaller data sets (intervals) of primes exhibit a clear bias in first digit distribution. The researchers noticed another pattern: the larger the data set of primes they analyzed, the more closely the first digit distribution approached uniformity. In light of this, the researchers wondered if there existed any pattern underlying the trend toward uniformity as the prime interval increases to infinity.

(...)

The set of all primes - like the set of all integers - is infinite. From a statistical point of view, one difficulty in this kind of analysis is deciding how to choose at “random” in an infinite data set. So a finite interval must be chosen, even if it is not possible to do so completely randomly in a way that satisfies the laws of probability. To overcome this point, the researchers decided to chose several intervals of the shape <1, 10^d>; for example, 1-100,000 for d = 5, etc. In these sets, all first digits are equally probable a priori. So if a pattern emerges in the first digit of primes in a set, it would reveal something about first digit distribution of primes, if only within that set.

By looking at multiple sets as d increases, Luque and Lacasa could investigate how the first digit distribution of primes changes as the data set increases. They found that primes follow a size-dependent Generalized Benford’s law (GBL). A GBL describes the first digit distribution of numbers in series that are generated by power law distributions, such as <1, 10^d>. As d increases, the first digit distribution of primes becomes more uniform, following a trend described by GBL. As Lacasa explained, both BL and GBL apply to many processes in nature.

http://www.physorg.com/news160994102.html

[ThirdWorldJohn]

1. Tim Allen And Tool Time Approves N/T

[Webster Green]

2. Thanks for reminding me how much I hate math.

I have absolutely no aptitude for it whatsoever. It actually makes me dizzy. :P

[Towlie]

3. Brain hurt.

[hvn_nbr_2]

4. Do these principles of first digits apply if using a base other 10?

Not that I expect you to know that offhand. But it seems to me like a reasonable question to ask, since the choice of base 10 seems rather arbitrary. If three-toed sloths did mathematics, they'd probably use base 6, and I wonder if they'd find the same kinds of patterns.

[phantom power]

5. Yes, in fact they would apply

It's a consequence of the logarithmic nature of the numbering system. That is true regardless of the base.

[hvn_nbr_2]

6. Thanks.

I suspected so but didn't think it through.

[eppur_se_muova]

7. This sounds like something that should follow directly from the Prime Number Theorem ...

http://primes.utm.edu/glossary/xpage/PrimeNumberThm.html

and the form of any base-n power series representation.

Not expert enough to provide a proof, but when you bear in mind that there more primes in the interval (10000,19999] than in (90000,99999], and that all of the former start with the digit 1 while all of the latter start with 9, that should cover it. Of course, if you are examining the interval from zero, say (0,100000], then you have to include comparison of the intervals (10,19],(100,199] ... to (90,99],(900,999], etc., but there are so many more numbers in the larger intervals that their effects should dominate. The smaller intervals will follow the law in the same general fashion, just not precisely the same.

It wouldn't surprise me if the PNT plus some algebra led to Benford's Law directly.

(I tend to doubt the meaningfulness of a theorem which assumes a particular base aka radix. Only base 2, the smallest possible integer base for a power series representation, is really unique.)