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## New Pattern Found In Prime Numbers509

stephen.schaubach writes "Spanish Mathematicians have discovered a new pattern in primes that surprisingly has 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. ... Besides providing insight into the nature of primes, the finding could also have applications in areas such as fraud detection and stock market analysis. ... 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.'"
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## New Pattern Found In Prime Numbers

• #### Other bases? (Score:5, Insightful)

on Sunday May 10, 2009 @10:38AM (#27896535)

When happens with the primes are represented in base-9 or base-11?

• #### Re:Other bases? (Score:5, Funny)

by Anonymous Coward on Sunday May 10, 2009 @10:41AM (#27896559)

• #### Re:Other bases? (Score:5, Funny)

by Anonymous Coward on Sunday May 10, 2009 @10:47AM (#27896631)

• #### Re:Other bases? (Score:5, Funny)

<imipak@yah[ ]com ['oo.' in gap]> on Sunday May 10, 2009 @02:20PM (#27898345) Homepage Journal

"Bad" as in you will see the Message as hinted at by Carl Sagan's "Contact". It's from God and apparently decodes to: "We apologize for the inconvenience".

• #### Re:Other bases? (Score:5, Funny)

on Sunday May 10, 2009 @03:13PM (#27898753)
• #### Re:Other bases? (Score:4, Funny)

by Anonymous Coward on Sunday May 10, 2009 @04:40PM (#27899351)

AND end with 1...this must be a conspiracy

• #### Re:Other bases? (Score:5, Informative)

on Sunday May 10, 2009 @10:46AM (#27896621)

Benson's Law is actually independent of the number base used. It wouldn't be much of a mathematical property if it wasn't. No matter how you convert a number, you will always see the same bias.

• #### Re:Other bases? (Score:5, Insightful)

<`jonaskoelker' `at' `yahoo.com'> on Sunday May 10, 2009 @10:54AM (#27896689)

I don't know; it might be interesting to know that the leading digits of powers-of-k are distributed in some interesting way in base not-k. They obviously all have a leading 1 in base k.

• #### Re:Other bases? (Score:5, Informative)

by Anonymous Coward on Sunday May 10, 2009 @01:18PM (#27897777)

They are also distributed as Benson's law describes, providing that k is not a rational power of the base. IAAM.

• #### Re:Other bases? (Score:5, Funny)

on Sunday May 10, 2009 @10:31PM (#27901639) Journal

IAAM.

Wow, first use of "I am a moron" I've seen in the field!

Hmm, or it is Mormon?

• #### Re:Other bases? (Score:4, Interesting)

by Anonymous Coward on Sunday May 10, 2009 @10:59AM (#27896749)

Benson's Law is actually independent of the number base used. It wouldn't be much of a mathematical property if it wasn't.

Err, what? The study of representations of numbers is a valid field of mathematics itself.

• #### Re:Other bases? (Score:5, Informative)

by Anonymous Coward on Sunday May 10, 2009 @12:08PM (#27897237)

Numbers are objects, I wish people would understand that numbers are just distinctions. The whole of mathematics is really just a language of form and structure, a system to systematize and decribe structure and forms (relationships are a type of form).

• #### Re:Other bases? (Score:5, Interesting)

on Sunday May 10, 2009 @10:43PM (#27901701)

The whole of mathematics is really just a language of form and structure, a system to systematize and decribe structure and forms (relationships are a type of form).

So... mathematics is the vaguest thing possible?

• #### Re: (Score:3, Insightful)

You should see Category Theory, "Well, a morphism is when you've got some things, and then you end up with some stuff...."
• #### Re:Other bases? (Score:5, Funny)

on Sunday May 10, 2009 @11:23AM (#27896901) Homepage

• #### Re:Other bases? (Score:5, Insightful)

on Sunday May 10, 2009 @11:39AM (#27897003)

...and all but one would end with 1 as well.

• #### Re:Other bases? (Score:4, Interesting)

by Anonymous Coward on Sunday May 10, 2009 @12:00PM (#27897167)

But how many would contain all 1s? Answer that, and provide a proof for your answer, and you'll make math history.

• #### Re:Other bases? (Score:5, Informative)

on Sunday May 10, 2009 @01:42PM (#27897999) Homepage

You mean, how many are Mercene primes [wikipedia.org]?

• #### Re:Other bases? (Score:5, Funny)

by Anonymous Coward on Sunday May 10, 2009 @04:04PM (#27899091)

I will never understand how people do that. You have the link right there. Even if you didn't open it to make sure, the link itself mentions the name "Mersenne Prime", and yet you write Mercene.

• #### Re: (Score:3, Interesting)

But how many would contain all 1s? Answer that, and provide a proof for your answer, and you'll make math history.

Obviously the number of digits would have to be a prime number. But not all prime number of digits would give you a prime number. The first case is when there are 11 digits, the number would be 23*89 in that case.

• #### Re:Other bases? (Score:5, Informative)

<Simetrical+sd@gmail.com> on Sunday May 10, 2009 @03:13PM (#27898757) Homepage

But how many would contain all 1s? Answer that, and provide a proof for your answer, and you'll make math history.

For those who didn't get it: it's not known whether there are infinitely many Mersenne primes [wikipedia.org], which have this form in binary (they're primes of the form 2^n - 1). Similarly, if you could figure out how many primes have only their first and last bits equal to 1, you would answer a longstanding question about Fermat primes [wikipedia.org] (which are primes of the form 2^n + 1).

• #### Re: (Score:3, Interesting)

The Wikipedia article states that in practice more than the first digit is used.

Is this something like a histogram, and increasing the number of digits analyzed gives you a better picture?

• #### Re: (Score:3, Informative)

Right, but the law is regarding distribution among the digits of possible first digits. Given that 1 is the only possible first digit in base 2, it could still be said to hold.

I don't actually know, I'm guessing here. But it seems like in base-8, you wouldn't be looking to include the digit 9 in your distribution analysis.

• #### Re:Other bases? (Score:5, Funny)

<(auxiliary.addre ... (at) (gmail.com)> on Sunday May 10, 2009 @02:08PM (#27898241)

100% = 100/100 = 1 = 0b1, which, by the way, looks like "Obi" and sounds like "Obi-Wan" when you say it.

• #### Re:Other bases? (Score:4, Interesting)

on Sunday May 10, 2009 @11:31AM (#27896961)
Code this have cryptographical uses? IANAMOG, but I know primes play a role in many crypto schemes.
• #### Re:Other bases? (Score:5, Funny)

on Sunday May 10, 2009 @01:15PM (#27897751)
Some encryption algorithms that were predicted to take forever to crack with today's technology, may in the long run end up taking the logarithm of forever.
• #### Re: (Score:3, Interesting)

Since "forever" is often exponential time, the logarithm of forever would indeed make cryptanalysis easy. :)
• #### Re: (Score:3, Interesting)

Since RSA relies on it being a Hard Problem to factor a number that is the product of two very large primes, it potentially introduces a weakness, as it presumably means some of those products will be easier to factor than others. A number that can be shown to be the product of two primes that both start with 9 should be much easier to work on than a number where both primes start with a 1, as there are far fewer 9- primes and therefore a smaller search space.

The obvious place to start looking would be the

• #### Re:Other bases? (Score:5, Funny)

on Sunday May 10, 2009 @10:54AM (#27896697) Homepage

base-9 or base-11?

NEVER FORGET

• #### Re:Other bases? (Score:5, Informative)

on Sunday May 10, 2009 @10:55AM (#27896699)
It wouldn't change the logarithmic nature of the distribution of the digits, AFAIK.

My math degree is getting dusty, but I'm pretty sure that the same pattern could be represented in another base by changing their generalization of Benford's law to include it, and the distribution would look like log(x)/log(9) or log(x)/log(11). Remember, changing the base of a logarithm is easy: for example, log(x)/log(e) = ln(x)

So you get the same distribution, different base.
• #### Re: (Score:3, Interesting)

If you got more numbers in 10-19 than 90-99, you probably also have more numbers in 0x10-0x1f than 0xf0-0xff...

• #### Re: (Score:2, Interesting)

by Anonymous Coward

(Warning, IANAM)
It's base independent. Basically, primes are distributed on a logarithmic scale (prime number theorem). For sufficently large intervals, there are always more primes in interval starting at x than the interval of the same size starting at y if xy. Like, there are more prime numbers in 1000..1999 than in 9000..9999.

• #### Re:Other bases? (Score:5, Informative)

on Sunday May 10, 2009 @11:07AM (#27896797) Journal

"The result holds regardless of the base in which the numbers are expressed, although the exact proportions change."

• #### Re:Other bases? (Score:5, Funny)

by Anonymous Coward on Sunday May 10, 2009 @12:55PM (#27897599)

Oh yeah? Well give me two minutes and check again.

• #### Re: (Score:3, Funny)

I just did it in base-2 and found that 100% of all primes start with the digit 1.

• #### Re:Other bases? (Score:5, Funny)

on Sunday May 10, 2009 @11:10AM (#27896819)

All your base are belong to Benford.

• #### Re:Other bases? (Score:5, Funny)

on Sunday May 10, 2009 @04:13PM (#27899155) Homepage
You have no chance to survive make your prime.
• #### Re: (Score:2)

I've often wondered how many patterns we are missing, especially in regards to our "special" numbers (Pi, Phi, e, primes....) because we mostly deal in base10. If we suffer a Class 1 or Class 2 Apocalypse [tvtropes.org], I intend to seize the opportunity and implement hexadecimal as a default number system.
• #### Re: (Score:2)

obxkcd:

http://xkcd.com/567/ [xkcd.com]

• #### Re:Other bases? (Score:5, Informative)

on Sunday May 10, 2009 @12:15PM (#27897309)

Benford's law works by the observation that, when numbers come up in certain real world contexts, the fluctuations you get in numbers should be proportional to the numbers themselves. Phrased differently, variations tend to be relative, not absolute. Because of this, if you have a very large range of random numbers from many real world measurements, then you would expect the number between t and t*(1.0001) not to vary too much for small changes in t. Let us try to use this observation very coarsely. Among the numbers with 6 digits, the number that look like 1xxxxxx (those between 100000 and 200000) should be about the same the number between 200000 and 400000. The same thing happens with the numbers with 5 digits or 7 digits or n digits (assuming that you have a wide range of random numbers, and the numbers are the kind that come from certain sorts of real world measurements). Additionally, you can get distributions for the first two digits, the first three digits, etc.

This observation doesn't depend on the base that you're working with.

Now, with the prime numbers, they have a distribution that is different from a lot of real world measurement data. The number of primes between n and n+d is approximately d/ln(n), where ln is the log with base e and d is small compared to n. So the number of primes between 500000 and 600000 is about 100000/ln(500000), and the number of primes between 500000 and 600000 is about 100000/ln(600000). By using this, and being slightly more careful, one can determine fairly easily the distribution of the leading terms of the prime numbers.

This is not a hard result. I would say that any professional mathematician who knew about the basic distribution of the primes could derive the distribution of the leading digis of the prime numbers fairly easily if anybody actually asked them to. The reason nobody mentioned this before is that nobody actually cares. While Benford's law does have applications to fraud detection, this new result does not. It's one of those things that makes people say "ooh, a pattern!" but which is just an easy and somewhat mundane corollary to a well known theorem.

• #### Duh (Score:3, Insightful)

by Anonymous Coward on Sunday May 10, 2009 @10:41AM (#27896567)

Benford's "law" is not a law at all... any exponential distribution will exhibit this behavior.

• #### Re: (Score:2, Funny)

by Anonymous Coward

You're right! I'm writing to my congress asking them to repeal Benford's Law.

• #### Re: (Score:3, Insightful)

Benford's "law" is not a law at all... any exponential distribution will exhibit this behavior.

A law, as the word is commonly used in math and physics, is a mathematical expression of a universal relationship. As you say, Benson's law is a property of any exponential distribution, so we agree it's universal. Why then can't we call it a law? Just because it's obvious after you understand it doesn't make it any less a law.

-JS

• #### Like batting orders (Score:4, Interesting)

<skh2003@columbia . e du> on Sunday May 10, 2009 @10:46AM (#27896617) Homepage Journal
I'm not a mathematician, could someone explain why this is surprising in terms that a computer programmer or biologist could follow? First thing I thought - no matter how many innings you have, you can guarantee that the top of the order will be up at least as many times as the bottom of the order.

Okay, if you have a random number along the interval (1,10^X), all the leading digits will be equally likely.

If you have some other interval (1,n*10^X), 1<=n<=9, then the leading digits > n will appear roughly 1/10 as often as leading digits 1..n.

If you have a large sample which is drawn from an admixture of some huge number of random distributions (1,n*10^X), with the "n" of each sub-distribution evenly distributed on 1..9, then the lower leading digits will be moderately more common, yeah?

Prime numbers, meanwhile, become decreasingly common as you get larger and larger, is that not correct? So it seems to me this is the obvious way to model prime numbers, no?
• #### On the density of prime numbers (Score:5, Insightful)

<`jonaskoelker' `at' `yahoo.com'> on Sunday May 10, 2009 @11:05AM (#27896779)

Prime numbers, meanwhile, become decreasingly common as you get larger and larger, is that not correct?

Yes, that is correct. There are roughly logarithmically many of them.

Bertrand's Conjecture (proven by Chebyshev) states than for all n > 1, there's a prime p with n < p < 2n.

If you look only at powers of two, it's readily seen that there are n primes between 1 and 2^n; setting k=2^n, there are log(k) primes between 1 and k.

A logarithmic upper bound follows from the Prime Number Theorem, which doesn't have an easy proof (AFAIK). It says something much more specific than just "It's O(log n)", though. Maybe there's a simple theorem from which you can derive O(log n), but I don't know.

• #### Re:On the density of prime numbers (Score:5, Informative)

<`jonaskoelker' `at' `yahoo.com'> on Sunday May 10, 2009 @11:16AM (#27896857)

Maybe if I had read the prime number theorem, I would have known that it's O(n / log n), which is somewhat bigger...

• #### Re: (Score:2, Interesting)

Good call.

Look at this: http://www.chaos.org.uk/~eddy/math/Benford.html [chaos.org.uk]

• #### Stock market analysis? (Score:5, Interesting)

on Sunday May 10, 2009 @10:48AM (#27896635)
I am admittedly not a mathematician, but I do have a good understanding of economics and finance, and I am not seeing how a pattern found in prime numbers could have any application to stock market analysis. Where is the interaction between prime numbers and the praxeology of buying and selling securities? Even if you're only focusing on automated buying and selling, those algorithms were still programmed by humans with their own subjective approaches and underlying premises.
• #### Re: (Score:2)

it makes more sense to me if you read that benfords law had applications in stock markets, fraud etc, but that isnt news...
• #### Re: (Score:2, Informative)

Benford's law can be used to detect fraud (the article states this, I don't have any reason to doubt it). They studied primes and found a pattern that is associated with a related property that they are calling Generalized Benford's Law. Presumably, the generalized rule can be used to detect a wider range of unnatural activity than Benford's law itself.

• #### Re:Stock market analysis? (Score:5, Funny)

on Sunday May 10, 2009 @11:22AM (#27896893) Homepage

I've always wondering how I could figure out when someone was trying to pass off a list of fraudulent primes. Glad to see that this problem is finally solved!

• #### Re:Stock market analysis? (Score:5, Interesting)

on Sunday May 10, 2009 @11:37AM (#27896995) Homepage Journal

I've always wondering how I could figure out when someone was trying to pass off a list of fraudulent primes. Glad to see that this problem is finally solved!

You're jesting, but I imagine that many fields of encryption would benefit from this, like dual key encryption, where the security lies in the ability to trust that the product really is of two primes, and that factoring this would be extremely time consuming.

Sets with a backdoor inserted may indeed have a different signature, and to be able to quickly see that one set differs would be invaluable. It wouldn't prove anything, but if, say, keys received from a certain company's key generator stood out like a sore thumb in a Benford distribution check, you would have reason to suspect foul play, incompetence or both.

• #### Re:Stock market analysis? (Score:5, Funny)

on Sunday May 10, 2009 @11:35AM (#27896981) Homepage Journal

I am admittedly not a mathematician, but I do have a good understanding of economics and finance, and I am not seeing how a pattern found in prime numbers could have any application to stock market analysis. Where is the interaction between prime numbers and the praxeology of buying and selling securities?

By understanding the patterns in prime numbers you can learn to spot them and avoid the sub-prime mortgage backed securities. Duh.

• #### The real article, and what it does and doesn't say (Score:3, Informative)

<`jonaskoelker' `at' `yahoo.com'> on Sunday May 10, 2009 @10:52AM (#27896667)

You can find the mathematicians' article at http://www.citeulike.org/group/3214/article/3664693 [citeulike.org] or http://arxiv.org/pdf/0811.3302 [arxiv.org] (pdf warning).

I find it interesting that the article doesn't prove any theorems. At least searching for the word "theorem" in the pdf only gives references to other theorems. Searching for "proof" gives no hits.

That leaves me thinking: what does this article tell us that we couldn't find out ourselves by ripping through some prime numbers? I thought the real power of math was to say something 100% certain about some infinitude of stuff, so we don't have to go and check every case by hand.

Oh well, I guess every open question needs some results on the form "this holds for all n <= bignum"; say, like the Goldbach Conjecture (every even number n > 2 is the sum of two primes).

• #### Re:The real article, and what it does and doesn't (Score:4, Insightful)

on Sunday May 10, 2009 @11:10AM (#27896821)

> That leaves me thinking: what does this article tell us that we couldn't find out ourselves by ripping through some prime numbers?

Nothing?

The important thing is that they ripped through some prime numbers and did notice, and they were the first to publish what they noticed.

The world moves forward in tiny steps like this. Maybe the next mathematician gets his 'Ahuh' moment on the back of an insight like this and bang modern crypto is fucked. He might even be able to prove it for you.

--Q

• #### Re:The real article, and what it does and doesn't (Score:2)

Their experimental result is a trivial consequence of the fact that prime number density around n is about 1/log(n). One could work out the exact theoretical distribution in one paragraph and that'd be all. I guess the authors are either ignorant or they prefer to market their result as "mysterious". Probably both.

• #### Re: (Score:2)

OK, not the most exciting science story of all time. Perhaps Carl Sagan either implanted or discovered a potential capacity for fascination with the science of primes in his novel "Contact" where a large sequence of prime numbers is used as an attempt by extra terrestrials to communicate with humanity.
• #### Cryptography? (Score:5, Funny)

on Sunday May 10, 2009 @10:55AM (#27896709)

Could this have any applications there?

"Well, I wasn't expecting The Spanish Mathematician . . ."

• #### Re:Cryptography? (Score:5, Funny)

on Sunday May 10, 2009 @11:06AM (#27896781)

Our two main powers are insight into the nature of primes, fraud detection
and stock market analysis.
I'll come in again...

• #### Good for them (Score:4, Funny)

on Sunday May 10, 2009 @10:56AM (#27896715)

Nobody expects the Spanish Mathematicians!

• #### What the arXiv paper says (Score:2, Informative)

by Anonymous Coward

Before I begin, I am a math phd candidate, but not in number theory. The following is probably better than a lay interpretation, but not an expert either.

Basically, they have generalized BL (Benson's Law) to get a GBL. They then tested the primes in the range [1,10^11] against GBL, and verified they were satisfied. They DID NOT PROVE THIS HOLDS FOR ALL PRIMES!!! They then went on to conjecture the applications of this to other areas (finance, etc).

Though the result is interesting, I really see this pape

• #### Stat Foop (Score:2)

Yeah, it only looks like that because started finding primes from 1 up. If we started finding them from infinity down...
• #### If you're dealing with phone numbers (Score:5, Interesting)

on Sunday May 10, 2009 @11:37AM (#27896991) Journal
It has less to do with math and more to do wit physics: as in how to use a an old school phone. Phone numbers, until comparatively recently would "prefer" lower numbers because they are EASIER TO DIAL. If a company had the phone number (909)999-9009 you would HATE dialing that thing. It would take about half a minute just to dial the damn number.

Ssssshhhhhhik!

Total pain in the finger.

1 as a first number was reserved for "other stuff" like international calls, so the lowest possible area codes (first numbers) went to places like New York City (212 - very quick to dial) or LA (213) because millions of people would be dialing that number, so it made for an overall faster dialing experience for (on average) more people.

This is compared to the relatively few people who lived in more obscure parts of the country, like Saginaw MI (989) or Bryan TX (979).

So, you have millions of phones in 212, thousands in 979. The result: saved effort in dialing.

Also, to this end there was a preference for exchanges to have lower numbers as well to save on dialing effort, and phone numbers with lower (but NON-ZERO) values were sought after. You'd see advertisments like "Call RotoRooter - 213 464 1111 !" or "Call us NOW for a free analysis! 201 738 1122 !" etc. and so on.

So, lower numbers in phone numbers have been a product of primitive dialing technology. Now with touchtone - all that is out the window - but the historic trend is still there and quite powerful - people will pay good money for a 212 area code for the distinction of being in the "real" New York Area code...

RS

• #### Re: (Score:3, Insightful)

Where are my mod points when I need them, that's pretty damned interesting.

• #### Re:If you're dealing with phone numbers (Score:5, Informative)

on Sunday May 10, 2009 @12:21PM (#27897379)
While you're absolutely right about the reasoning behind NYC, LA, and Chicago getting 212, 213, and 312, you're a little off on the 989 and 979 area codes, which are much more recent.

In the original system design, all area codes had a middle digit of 0 or 1. The convention was that a middle digit of 1 was used for area codes that only covered part of a state, while a middle digit of 0 was used for area codes that covered entire states. Furthermore, an area code could not begin with a 1 or a 0. and an area code with a middle digit of 1 couldn't have 1 as the third digit. (This left the shortest dial time area code for a statewide code as 201, which went to New Jersey.)

As early as the late 1950s, the idea of single area codes for some states went out the window (with NJ splitting into 201 and 609 in 1958) because of increasing population and proliferation of phone service.

By the late 1980s, the rules were further changed to allow for area codes with middle digits other than 1 or 0. Area codes like 989 and 979 weren't introduced until the late 1980s at the very earliest, by which point very few people were still using rotary phones. At one point, I had heard that the middle digit value of 9 was reserved for the future to allow for four digit area codes, but I can't vouch for the accuracy of that recollection. There are plenty of other rules, some of which you can see summarized here... [wikipedia.org]
-JMP
• #### Re:If you're dealing with phone numbers (Score:4, Informative)

on Sunday May 10, 2009 @01:26PM (#27897855)

So, you have millions of phones in 212, thousands in 979. The result: saved effort in dialing.

Nice idea, but you give the phone company too much credit. In the old days telephone switches still used physical relays (this is well before transistors were invented). This significantly limited the number of connections in progress each switch could handle. Since switches are expensive, you naturally wanted to pass on the call as fast as possible so you could free up the switch for the next caller. A number like '212' wasn't just easy to dial, it was fast — remember this is the era of pulse dialing as well, so a '9' took literally 9 times longer to dial than a '1'. Assigning fast numbers like '212' to New York saved money for the phone company because Ma Bell could buy fewer switches. Any benefit to the customer was purely accidental.

-JS

• #### Independent Verification (Score:5, Interesting)

<eldavojohn@ g m a i l .com> on Sunday May 10, 2009 @11:43AM (#27897031) Journal
Here's what I got on my own counts using the first million primes [utm.edu]:

1: 415441
2: 77025
3: 75290
4: 74114
5: 72951
6: 72257
7: 71564
8: 71038
9: 70320

Which puts the probabilities at:

1: 0.415441
2: 0.077025
3: 0.07529
4: 0.074114
5: 0.072951
6: 0.072257
7: 0.071564
8: 0.071038
9: 0.07032

My computer is currently crunching the first fifty million primes and I will post those as a reply to this post later today when it is done.

These ratios on numbers 2-9 seem far too close in range for this to be a true log scale. Hopefully with more data it will be more logarithmic.

• #### Re:Independent Verification (Score:5, Funny)

on Sunday May 10, 2009 @11:55AM (#27897121) Journal

This is one of those moments that I love /.
Personally, I was trying to calculate the first 50M primes using the sieve of Erastothenes and then contructing a program that categorizes them but since you are doing all the work I say go ahead and I'll wait for the results.

• #### I Found a Fit! (Score:5, Interesting)

<eldavojohn@ g m a i l .com> on Sunday May 10, 2009 @06:57PM (#27900261) Journal
The results for all primes between one and one hundred million:

Counted Occurances:
686048, 664277, 651085, 641594, 633932, 628206, 622882, 618610, 614821
Frequencies:
0.119, 0.115, 0.113, 0.111, 0.110, 0.109, 0.108, 0.107, 0.107

So I hope that satisfies everyone who replied to my thread first of all. I hope 5,761,455 primes between one and one hundred million satisfies you.

I used a very simple Non Linear Squares model to solve for a single constant on a log of these values. I think I have a fit. Using Benford's model and the NLS Package in R, I found:

f(x) = 0.020814 * log(161.147689 * ((x+1)/x))

To fit quite nicely, here's the summary:

Formula: y ~ Const1 * log(Const2 * ((x + 1)/x))

Parameters:
Estimate Std. Error t value Pr(>|t|)
Const1 0.020814 0.001940 10.7292 1.343e-05 ***
Const2 161.147689 80.222081 2.0088 0.08452 .
---

Residual standard error: 0.0010413 on 7 degrees of freedom

Number of iterations to convergence: 8
Achieved convergence tolerance: 1.8104e-07

Here is the list of frequencies next to what my model produced:

Benford Prime Rates
0.11907548
0.11529674
0.11300704
0.11135972
0.11002984
0.10903600
0.10811193
0.10737045
0.10671280

NLS Model Results
0.1202106
0.11422279
0.11177125
0.11042794
0.10957828
0.10899193
0.10856276
0.10823497
0.10797641

I would wager that they are correct. Neat discovery!

• #### Enron (Score:5, Interesting)

on Sunday May 10, 2009 @11:45AM (#27897059) Journal

was busted by auditors who found the books were "cooked" by applying the law of first numbers described in the /. blurb and TFA. The independent auditors found the first figures were randomly distributed instead of following Benford's law with the number 1 the most plentiful and nine the least -- therefore, the entries were fraudulent.

Benford's law knocked my out at the time; I thought of how many bogus figures I had entered in my expense accounts over the years....

• #### Re:Enron (Score:4, Informative)

on Sunday May 10, 2009 @12:42PM (#27897529) Homepage Journal

At smaller scales than Enron, Benford and other related number distribution analysis schemes are indeed used to find fraud.

Know the time report you fill out for the company you work at? There's a chance that it passes through a filter. If you make up figures for how long you spend at certain tasks, someone higher up may see it with a footnote saying "High probability of data being fictitious". If this pattern repeat itself over months, don't be surprised if your chances of retaining your job diminishes.

On the other hand, you can use the statistics for your own advantage too. Not the least in games and gambling, where guessing your competitor's status and actions can change the odds quite dramatically. Including games and gambling you don't think of as games and gambling, like placing bids in auctions.

• #### Re:Enron (Score:4, Interesting)

<pkteisonNO@SPAMhotmail.com> on Sunday May 10, 2009 @05:22PM (#27899631) Homepage

Great example. Here's a pretty good article on it: http://abcnews.go.com/print?id=98043 [go.com]

I also like their explanation for "why":
"Imagine that you deposit \$1,000 in a bank at 10 percent compound interest per year. Next year you'll have \$1,100, the year after that \$1,210, then \$1,331, and so on. The first digit of your bank account remains a "1" for a long time.

When your account grows to more than \$2,000, the first digit will remain a "2" for a shorter period as your interest increases. And when your deposit finally grows to more than \$9,000, the 10 percent growth will result in more than \$10,000 in your account the following year and a long return to "1" as the first digit. "

• #### Not surprising at all (Score:4, Informative)

on Sunday May 10, 2009 @12:15PM (#27897319)

If you had asked me about the distribution of first digits of prime #s yesterday I probably would have guessed logarithmic, regardless of base (except for binary, of course).

Think about it. We know that primary # are distributed logarithmically. A set of N digit #s has equal subsets of numbers starting with 1, 2, 3, etc. Those subsets are equal in size, exclusive and completely ordered with respect to each other. So it follows that the # of prime #s in consecutive subsets would be a logarithmic function. And if you add the sizes of prime subsets for each starting digit, you'll still get a logarithmic distribution.

Nothing to see here, move along.

m

• #### Not predictive (just in case you were thinking it) (Score:4, Interesting)

on Sunday May 10, 2009 @12:59PM (#27897625) Homepage

If this is the first time you've run across Benford's law, you might thought to yourself, "Wow, I can use that to predict large prime numbers! I'll just convert numbers to base X, where X is really big, and only check numbers that start with 1."

It's worth actually trying this, if you get a minute. You'll find that as X gets large, the number of primes that start with 2 gets closer to the number of primes tat start with 1. As X approaches infinity, your distribution approaches a smooth logarithmic curve.

It's neat to see it yourself. This gives you an easy way to experiment with an infinite, easily generated set of numbers that follows Benford's law. It turns out that math actually works, oddly enough.

• #### Complete bullshit (Score:5, Insightful)

on Sunday May 10, 2009 @01:04PM (#27897663)
The prime number theorem was conjectured in 1796 by Adrien-Marie Legendre and proved in 1896 independently by Jacques Hadamard and Charles Jean de la Vallée-Poussin. It says that if pi (N) denotes the number of primes p = N, then pi (N) / (N / ln N) converges towards 1; accordingly the number of primes between A and B is about (B / ln B - A / ln A). This shows that there should be slightly more d digit primes starting with 1 than with 2, 3, 4 etc. A reasonably good approximation is that the number of d digit primes starting with 1 is not 1/9 th of all d digit primes, but more precisely (11 1/9 + 5.7 / d) percent. This is all very, very simple maths. I don't think it hasn't been observed before, it was just never considered worth mentioning. However, the prime number theorem alone is not enough to prove this; it would be necessary to prove that convergence happens at a certain speed. So anything that these so-called "mathematicians" claim that depends on observations of large list of primes is pure nonsense.
• #### Re: (Score:3, Interesting)

I was thinking this myself at first, but apparently something goes wrong with it, because (per TFA) the distribution actually moves away from Benford's Law weighting and toward uniformity as the sample size grows larger. The specific rate of this movement is somehow described by a generalization of Benford's Law; while BL at one point uses 1/x, GBL uses 1/(x^a), with BL as the special case where a = 1.

<eldavojohn@ g m a i l .com> on Sunday May 10, 2009 @04:39PM (#27899343) Journal
So I read the comments and see that I need to do this in ranges or 1 to 100, 1 to 1000, etc. Which is fine, I've added another R method and would post the code here if it didn't yell at me for junk characters. So here are your Benford lists:

All Primes 1-100
Counted Occurances:
4, 3, 3, 3, 3, 2, 4, 2, 1
Frequencies:
0.160, 0.120, 0.120, 0.120, 0.120, 0.080, 0.160, 0.080, 0.040

All Primes 1-1,000
Counted Occurances:
25, 19, 19, 20, 17, 18, 18, 17, 15
Frequencies:
0.149, 0.113, 0.113, 0.119, 0.101, 0.107, 0.107, 0.101, 0.089

All Primes 1-10,000
Counted Occurances:
160, 146, 139, 139, 131, 135, 125, 127, 127
Frequencies:
0.130, 0.119, 0.113, 0.113, 0.107, 0.110, 0.102, 0.103, 0.103

All Primes 1-100,000
Counted Occurances:
1193, 1129, 1097, 1069, 1055, 1013, 1027, 1003, 1006
Frequencies:
0.124, 0.118, 0.114, 0.111, 0.110, 0.106, 0.107, 0.105, 0.105

All Primes 1-1,000,000
Counted Occurances:
9585, 9142, 8960, 8747, 8615, 8458, 8435, 8326, 8230
Frequencies:
0.122, 0.116, 0.114, 0.111, 0.110, 0.108, 0.107, 0.106, 0.105

All Primes 1-10,000,000
Counted Occurances:
80020, 77025, 75290, 74114, 72951, 72257, 71564, 71038, 70320
Frequencies:
0.120, 0.116, 0.113, 0.112, 0.110, 0.109, 0.108, 0.107, 0.106

This is the raw data so to turn that into something visual, I dumped it into a Google spreadsheet and made it public [google.com] (note the scale on the y axis). Enjoy!

It seems that the curve is flattening out the more data I collect, but the logarithmic curve may be valid. I have the data for 100,000,000 and will add that to the spreadsheet once it completes.

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