May 8, 2009 feature

# New Pattern Found in Prime Numbers

(PhysOrg.com) -- Prime numbers have intrigued curious thinkers for centuries. On one hand, prime numbers seem to be randomly distributed among the natural numbers with no other law than that of chance. But on the other hand, the global distribution of primes reveals a remarkably smooth regularity. This combination of randomness and regularity has motivated researchers to search for patterns in the distribution of primes that may eventually shed light on their ultimate nature.

In a recent study, Bartolo Luque and Lucas Lacasa of the Universidad Politécnica de Madrid in Spain have discovered a new pattern in primes that has surprisingly gone unnoticed until now. They found that the distribution of the leading digit in the prime number sequence can be described by a generalization of Benford’s law. In addition, this same pattern also appears in another number sequence, that of the leading digits of nontrivial Riemann zeta zeros, which is known to be related to the distribution of primes. Besides providing insight into the nature of primes, the finding could also have applications in areas such as fraud detection and stock market analysis.

“Mathematicians have studied prime numbers for centuries,” Lacasa told *PhysOrg.com*. “New insights and concepts coming from nonlinear science, such as multiplicative processes, help us to look at prime numbers from a different perspective. According to this focus, it becomes significant that even today it is still possible to discover unnoticed hints of statistical regularity in such sequences, without being an expert in number theory. However, the most significant issue in this work is not to unveil this pattern in primes and Riemann zeros, but to understand the reason and implications of such unexpected structure, not just for number theoretical issues but, interestingly, for other disciplines as well. For instance, these results deepen our understanding of correlations in systems composed of many elements.”

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.

“Imagine that you have $1,000 in your bank account, with an interest rate of 1% per month,” Lacasa said. “The first month, your money will become $1,000*1.01 = $1,010. The next month, $1,010*1.01, and so on. After n months, you will have $1,000*(1.01)^n. Notice that you will need many months to go from $1,000 to $2,000, while to go from $8,000 to $9,000 will be much easier. When you analyze your accounting data, you will realize that the first digit 1 is more represented than 8 or 9, precisely as Benford's law dictates. This is a very basic example of a multiplicative process where 0.01 is the multiplicative constant.

“Physicists have shown that many processes in nature can be modeled as stochastic multiplicative processes, where the previously constant value of 0.01 is now a random variable and the data equivalent to the money of our latter example is another random variable with an underlying distribution 1/x. Stochastic processes with such distributions are shown to follow BL. Now, many other phenomena fit better to a stochastic process with a more general underlying probability x^[-alpha], where alpha is different from one. The first digit distribution related with this general power law distribution is the so-called Generalized Benford law (which converges to BL for alpha = 1).”

Significantly, Luque and Lacasa showed in their study that GBL can be explained by the prime number theorem; specifically, the shape of the mean local density of the sequences is responsible for the pattern. The researchers also developed a mathematical framework that provides conditions for any distribution to conform to a GBL. The conditions build on previous research, which has shown that Benford behavior could occur when a distribution follows BL for particular values of its parameters, as in the case of primes. Luque and Lacasa also investigated the sequence of nontrivial Riemann zeta zeros, which are related to the distribution of primes, and whose distribution of the zeros is considered to be one of the most important unsolved mathematical problems. Although the distribution of the zeros does not follow BL, here the researchers found that it does follow a size-dependent GBL, as in the case of the primes.

The researchers suggest that this work could have several applications, such as identifying other sequences that aren’t Benford distributed, but may be GBL. In addition, many applications that have been developed for Benford’s law could eventually be generalized to the wider context of the Generalized Benford’s law. One such application is fraud detection: while naturally generated data obey Benford’s law, randomly guessed (fraudulent) data do not, in general.

“BL is a specific case of GBL,” Lacasa explained. “Many processes in nature can be fitted to a GBL with alpha = 1, i.e. a BL. The hidden structure that Benford's law quantifies is lost when numbers are artificially modified: this is a principle for fraud detection in accounting, where the combinatorial mechanisms associated to accounting sets are such that BL applies. The same principle holds for processes following GBL with a generic alpha, where BL fails. Last, for processes whose underlying density is not x^(-alpha) but 1/logN, a size-dependent GBL would be the correct hallmark.”

__More information:__ Bartolo Luque and Lucas Lacasa. “The first digit frequencies of primes and Riemann zeta zeros.” *Proceedings of the Royal Society A*. doi: 10.1098/rspa.2009.0126.

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**Citation**: New Pattern Found in Prime Numbers (2009, May 8) retrieved 18 October 2019 from https://phys.org/news/2009-05-pattern-prime.html

## User comments

googleplexAshyNonRationaljselinUntil we know, we don't know. Primes are something so simple in concept yet we cannot predict them. The knowledge gained in doing so may shed light on things we cannot yet conceive of. Progress!

IcesterYes it matters. If we could find the true nature of primes and their distribution, then we can also do the same for non-primes (composites).

Now if someone can just figure out what good finding the composites are....

magpiesSMMAssociates:D

uberFubugmenot23BLKnightThis book is said to have discovered an easy way to calculate primes. McCanney says this discovery may weaken the backbone of Security Coding and Encryption, which is based on the characteristics of prime numbers.

This is not mentioned in this article. Perhaps the author is unawares of this book...

http://www.calcul...mes.com/

James A. Jancik

The "Black Knight" of Talk Radio

Feet to the Fire

JimsterAntStreetwisetkjtkjtell that to the physics instructor at Tufts Univ. who got the Nobel prize...without a phd

it does not seem that you even bothered to read his work before choosing to mouth-off!

mfritz0paulcurranJimster, can you prove this? Amazing at the level of ignorance being displayed by that comment, well more disappointed than amazed. There are many reasons for getting an MS over a PhD.

RodI don't know much about 'Prime Numbers' except what they are like divisible by only 1 and themselves like: 1,2,3,5,7,11,13, etc.

And that there is a formula for finding the 'Prime Numbers' between numbers 1 and 97. -After 97, the formula falls apart.

Maybe, someone, someday will complete the formula so that every 'Prime Number' can be found.

Though, Not really sure why we would want that formula, but anyway- its something to do.

ArthurDiColdDimSumYes they do. It's an abstract mathematical concept that doesn't depend on the representation.

markallumsmarkallumsj0equ1nnmatthew10Arguably the greatest threat our society and general status quo faces resides in our banking system. Right now fraud is being committed by the major banks and possibly even quasi governmental entitities (Federal Reserve). If and when people lose faith in the fiat currency systems currently in place, dangerous changes become quite possible - barter economies, political regime failures, etc. As such, rooting out fraud and exposing the bad actors becomes all the more necessary.

thalesJust as particle physicists work to understand the fundamental nature of matter, energy, and ultimately the universe through the basic building blocks of reality, so mathematicians working with primes want to understand the underlying fundamental nature of numbers.

This is particularly exciting when you consider that the rules of math may transcend this universe. We speak of the laws of nature, like gravity, inertia etc. but these are probably functions of the universe. The universe, however, may well be a function of math.

sjmillersjmillergoogleplexWhat about Einstein?

He did poorly at academics and was critical of its creativity stiffling structure - even failing a school entrance exam. Eventually at 26 he got his PhD but he struggled with the rigidity of academia. The PhD was given after his most important works.

laripuKRsixDDThere is a fascinating discussion related to this in "The Music of the Primes" by Marcus du Sautoy, pp.202-202, regarding how the Riemann conjecture would end up proved if someone found it is "undecidable".

KRsixDDMy typo, it's pp.201-202.

RicochetthalesUnfortunately, we'd have to grasp their significance before we could evolve into a pure energy-state. *sigh*

BTW, I don't think "magnamousness" is a word.

laripuOh, it _is_, but only for someone of Ricochet's vast ignonce. :-) That ignonce is well beyond human comprehension! :)

Linda95959By George, I think you have got it. Now what do we do with it? Just asking.

thalesWell I'm flattered that you think I would know the answer to that. Of course no one does really.

Having said that, the string theorists have a speculative but very mathematically rigorous approach that has had some apparent success answering these kinds of questions. Their models sound kooky, since most have 9 dimensions of space. "M theory", probably the most successful model, has 11 spatial dimensions. So they're counterintuitive, but they are mathematically consistent. If you want to know more, this is a pretty good primer:

http://www.supers...dex.html

Or you could always start with Wikipedia.

RicochetUhmm.. yeah... it's called "humor". Perhaps you've heard of it?

wordofmouthTaking that all numbers are infinite,

ultimately everything balances out,therefore any patterns are false and invalid in the proper scheme of infinity.

wordofmouththales- Principia Discordia

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