New Pattern Found in Prime Numbers

May 08, 2009 By Lisa Zyga feature
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, 10d]; 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, 10d]. 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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User comments : 41

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googleplex
3.7 / 5 (3) May 08, 2009
Intriguing. But this only pertains to the first digit. Perhaps they should normalize the numbers using an exponential base instead of 10.
Ashy
1 / 5 (1) May 08, 2009
At this form it can't be used for finding very long prime digits, but may be in future...
NonRational
1 / 5 (6) May 08, 2009
Does this matter, besides finding new primes for fun? What is to be gained by finding the "true nature" of primes? Fraud detection is mentioned, but that's not exciting (to me)
jselin
5 / 5 (7) May 08, 2009
Does this matter, besides finding new primes for fun? What is to be gained by finding the "true nature" of primes? Fraud detection is mentioned, but that's not exciting (to me)


Until 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!
Icester
4.5 / 5 (4) May 08, 2009
Does this matter, besides finding new primes for fun? What is to be gained by finding the "true nature" of primes? Fraud detection is mentioned, but that's not exciting (to me)


Yes 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....
magpies
1 / 5 (7) May 08, 2009
The true nature of primes is ez cmon ppl primes are old news...
SMMAssociates
4.7 / 5 (3) May 08, 2009
I understood most of that, but perhaps "42" is the correct answer....

:D
uberFu
not rated yet May 08, 2009
I have to agree with SMMAssociates 42 solves everything!
bugmenot23
5 / 5 (2) May 08, 2009
One practical application of prime numbers is computer security. Were a pattern identified to them, they would be much easier to generate, and therefore much easier to attack. Security would get magnitudes harder to maintain.
BLKnight
1 / 5 (2) May 09, 2009
"Calculating Primes" -James M. McCanney, M.S. Physics

This 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
Jimster
2.9 / 5 (8) May 09, 2009
"James M. McCanney, M.S. Physics" HA! Anybody who must settle for an MS rather than getting a PhD in Physics does so because they do not have the math skills ... and this guy write a book on how to "calculate" prime numbers! The irony is hilarious! BTW, one can not "calculate" prime numbers one can only search for them.
Ant
not rated yet May 09, 2009
No its not 42 its 21-22 interface thats the real answer
Streetwise
1 / 5 (2) May 09, 2009
Sorry, Jimster, but Mr. Jancik is spot on. You're quick to launch a personal attack on Jim McCanney but, like most debunkers, you don't do your homework. McCanney is a decade ahead of his contemporaries with groundbreaking works like "Calculate Primes." Read it. You might learn something. Part of McCanney's brilliance, also, is the ability to explain complex principles to the layman, a talent sorely lacking in scientific circles. While naysayers like you spend time huffing and puffing, Jim McCanney writes scholarly books and offers a weekly science podcast, at no cost to the public. Really, Jimster, what have YOU done for science lately? When you were in diapers, McCanney was lecturing at Cornell. For those seeking a better understanding of Earth's weather, I would highly recommend another McCanney book, "Principia Meteorologia: The Physics of Sun Earth Weather."
tkjtkj
5 / 5 (1) May 09, 2009
"Anybody who must settle for an MS rather than getting a PhD in Physics does so because they do not have the math skills"

tell 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!

mfritz0
not rated yet May 10, 2009
I believe there is a pattern to all things real and imaginary. It is through these real patterns that the human brain is capable of creative thought and perhaps, just perhaps, to opening all the secrets of the universe.
paulcurran
not rated yet May 10, 2009
"Anybody who must settle for an MS rather than getting a PhD in Physics does so because they do not have the math skills ... "

Jimster, 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.
Rod
5 / 5 (1) May 10, 2009

I 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.
ArthurDi
4.2 / 5 (5) May 10, 2009
I have discovered a remarkable method for quickly determining the primality of any integer, but unfortunately space limitations preclude me from including it in this comment.
MGraser
not rated yet May 10, 2009
I'm not a mathematician, as my comments will make clear. I think primes are tremendously interesting though, and I'm sure that understanding a pattern will reveal something about the nature of mathematics itself and thus the world we live in.

If you were dealing with a binary, hex, or other non-decimal based system, do primes exist? They are calculated so differently, that perhaps it's a non-entity in those systems? So, is there something that can be derived about prime numbers by looking at them through the lens of these other systems? Curious what anyone thinks about it.
ColdDimSum
5 / 5 (1) May 10, 2009
MGraser writes:
If you were dealing with a binary, hex, or other non-decimal based system, do primes exist?

Yes they do. It's an abstract mathematical concept that doesn't depend on the representation.
markallums
not rated yet May 10, 2009
Not 42, not 21:22 interface; obviously it's in the digits of transcendental numbers, especially PI.
markallums
not rated yet May 10, 2009
In base 11.
j0equ1nn
5 / 5 (4) May 11, 2009
I didn't read all the comments so I hope I'm not being redundant, but I know a bit about math and it looked like some people were looking for explanations of the importance of primes. First of all, they stand out to us on an intuitive level because of the unique property of not being divisible by anything. There is something very fundamentally significant about that. If you don't relate to the formal interest, it's hard to explain exactly "what we need them for" of "what you can do with them." Things generally work in the opposite order. We discover new mathematical ideas, then we discover that they lead to new scientific advances. For example, extensive formally-motivated studies of the complex numbers ended up providing models to support electric current flow, as well as aerodynamics. Of course, we didn't know this is where it would go when people were first asking questions about the square root of negative one, because those branches of science didn't exist yet. But I'm sure there were people then who said "who cares, it doesn't pertain to anything."
lengould100
2.3 / 5 (3) May 11, 2009
I can't think of any practical use of electricity which wouldn't have happened regardless of the development of formal analysis of complex numbers.... Simply because some mathematical concept is useful in describing a physical phenomenon doesn't mean that mathematics get full credit for the development of the applications of phenomenon.
matthew10
not rated yet May 11, 2009
I saw a couple of comments in re: what practical applicability this could have? Look no further than fraud identification. To those who say they have no interest in this field - I say stick your head back in the sand where it belongs. This affects us all.

Arguably 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.
thales
5 / 5 (1) May 11, 2009
The way I think of it is that prime numbers are the only fundamentally "real" numbers since all other numbers are made of them.

Just 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.
sjmiller
not rated yet May 11, 2009
For numbers like primes, the percentage with leading digit 1 depends greatly on where you truncate. If you look up to 10^k or up to 9.9999 * 10^k, you get a very different answer. One solution, related to a comment above, is to look at these numbers with analytic and not the natural density. Using these weights, it has long been known that Benford's law holds. See Serre's classic "A Course in Arithmetic", page 76 (where Serre refers to a comment by Bombieri).
sjmiller
not rated yet May 12, 2009
Whoops - typo in what I wrote (that's the problem in quickly posting when running to get the kids from daycare). It should read the answer looks very different if you go up to 2 * 10^k or up to 9.9999 * 10^k; in the first we've just gone through a long stretch of 1s, in the latter we've just gone through a long stretch with no 1s. The standard example is to look at the distribution of leading digits of the integers in [1,n] as a function of n; it oscillates between (roughly) 5/9 when you stop around 2 * 10^k and about 1/9 (when you stop around 9.9999 * 10^k), or something along these lines.
googleplex
not rated yet May 12, 2009
"James M. McCanney, M.S. Physics" HA! Anybody who must settle for an MS rather than getting a PhD in Physics does so because they do not have the math skills ...

What 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.


laripu
not rated yet May 12, 2009
The reason primes are interesting is that they are fundamental building blocks fo integers, just like atoms are fundamental building blocks for matter. In fact, since number theory can be used to describe formal systems (see Godel's work), primes can also be thought of as a basis for logic. (This is a reversal of the usual was of thinking, but that's all right :)
KRsixDD
not rated yet May 13, 2009
The reason primes are interesting is that they are fundamental building blocks fo integers, just like atoms are fundamental building blocks for matter. In fact, since number theory can be used to describe formal systems (see Godel's work), primes can also be thought of as a basis for logic. (This is a reversal of the usual was of thinking, but that's all right :)

There 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".
KRsixDD
not rated yet May 14, 2009
The reason primes are interesting is that they are fundamental building blocks fo integers, just like atoms are fundamental building blocks for matter. In fact, since number theory can be used to describe formal systems (see Godel's work), primes can also be thought of as a basis for logic. (This is a reversal of the usual was of thinking, but that's all right :)


There 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".

My typo, it's pp.201-202.
Ricochet
not rated yet May 14, 2009
Primes are very important. So much so that the sheer magnamousness of their importance is well beyond human comprehension. Seriously, we'd have to evolve into a pure energy-state in order to truly grasp their significance.
thales
not rated yet May 15, 2009
Primes are very important. So much so that the sheer magnamousness of their importance is well beyond human comprehension. Seriously, we'd have to evolve into a pure energy-state in order to truly grasp their significance.


Unfortunately, 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.
laripu
not rated yet May 15, 2009
Primes are very important. So much so that the sheer magnamousness of their importance is well beyond human comprehension. Seriously, we'd have to evolve into a pure energy-state in order to truly grasp their significance.


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

Oh, it _is_, but only for someone of Ricochet's vast ignonce. :-) That ignonce is well beyond human comprehension! :)
Linda95959
not rated yet May 17, 2009
The way I think of it is that prime numbers are the only fundamentally "real" numbers since all other numbers are made of them.



Just 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.


By George, I think you have got it. Now what do we do with it? Just asking.
thales
not rated yet May 18, 2009
By George, I think you have got it. Now what do we do with it? Just asking.


Well 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.
Ricochet
not rated yet May 22, 2009

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


Uhmm.. yeah... it's called "humor". Perhaps you've heard of it?
wordofmouth
not rated yet May 27, 2009
Intriguing. But this only pertains to the first digit. Perhaps they should normalize the numbers using an exponential base instead of 10.

Intriguing. But this only pertains to the first digit. Perhaps they should normalize the numbers using an exponential base instead of 10.

Intriguing. But this only pertains to the first digit. Perhaps they should normalize the numbers using an exponential base instead of 10.

Taking that all numbers are infinite,
ultimately everything balances out,therefore any patterns are false and invalid in the proper scheme of infinity.
wordofmouth
not rated yet May 27, 2009
Using the number 5 in all calculations,never deviating, will make things much easier to prove my point.
thales
not rated yet May 29, 2009
"Everything in the universe relates to the number 5, one way or another, given enough ingenuity on the part of the interpreter."

- Principia Discordia