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New Pattern Found In Prime Numbers
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.'"
When happens with the primes are represented in base-9 or base-11?
I am becoming gerund, destroyer of verbs.
Benford's "law" is not a law at all... any exponential distribution will exhibit this behavior.
Explain one man being hit seven times with lightning. http://en.wikipedia.org/wiki/Roy_Sullivan
Improbable doesn't mean impossible.
If sharing a song makes you a pirate, what do I have to share to be a ninja?
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?
The good and new comes from no quarter where it is looked for, and is always something different from what is expected.
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.
Slashdot: Playing Favorites Since 1997
You can find the mathematicians' article at http://www.citeulike.org/group/3214/article/3664693 or http://arxiv.org/pdf/0811.3302 (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).
The real question is did his feigned interest result in sexual intercourse?
Could this have any applications there?
"Well, I wasn't expecting The Spanish Mathematician . . ."
Schroedinger's Brexit: The UK is both in and out of the EU at the same time!
hello troll, your inability to understand mathematics does not mean it has no real world application. her little project may well have been able to provide the basis for some ecomonic or social model, or may proove vital in unlocking the bit of physics that enables the next revolution in technology. Besides all these very important uses that skip the average joe, mathematics is often elegant and beautiful, and may be considered a form of art by some people
Nobody expects the Spanish Mathematicians!
Perhaps she was wondering the same about you as you walked away looking dumbfounded.
Just because something is complicated and difficult for most people to grasp doesn't mean it hasn't got some real-world application at some point. That's why we need people like her to make sense of that sort of stuff, to the benefit of the rest of us.
Explain one man being hit seven times with lightning. http://en.wikipedia.org/wiki/Roy_Sullivan
Poor bastard. After the fourth! time he began carrying a pitcher of water with him... I find it hard not to be amused.
that makes my /. id even more impressive :)
factor 966971: 966971
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.
A few examples:
For the same reason some people take Philosophy, Ancient Literature, Paleontology, etc. Because they think the subject is cool, and aren't necessarily at school to learn a trade. (Indeed, Engineering students that are paying attention also discover they aren't directly being taught a trade either. Or at least they aren't in any Engineering college worthy of the name.)
They want to become an actuary. This is a fairly well-paid job that is also rather difficult to do, and even harder to do well.
They want to become math teachers; a valuable and much-needed profession. Math is a useful tool in teaching students how to think. We certainly don't torture legions of high school students with the details of conic sections because anybody is under the impression this is a directly practical skill for most citizens to have. Nor are hundreds of thousands of college students subjected to the horrors of calculus because of some kind of employment program for math post-docs.
They are double-majors in a field in which math is extremely important (physics, astronomy, computer science, every type of engineering, linguistics, medicine, biology, etc. Pretty much every field outside the humanities. Oh, and some of the humanities make extensive use of math too.)
SirWired
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 paper as a conjecture on the nature of primes, related to the Riemann zeta function. (From what I understand someone has proved zeros of the zeta function follow GBL.)
> 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
Ssssshhhhhhik!
diggadiggadiggadiggadiggadiggadiggadiggadigga!
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
Shoes for Industry. Shoes for the Dead.
Which puts the probabilities at:
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.
My work here is dung.
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....
They found that the distribution of the leading digit in the prime number sequence can be described by a generalization of Benford's law.
Yeah, how did we miss that? We need to pay more attention.
To prevent this day from getting worse, I'll just read ERROR as GOOD TH
This reminds me of an interesting article (PDF) I found a while back which explains Benford's law nicely. To quote:
Proud member of the Ferengi Socialist Party.
Or (to put very crudely in layman-like terms): In a world where billions of things happen every single day, "1 in a million" events happen all the time.
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
...good god my sarcasm detector failed, I deserve a mod down for that one.
My webcomic
Mathematicians and mathematical results are generally indifferent to base. The mathematical properties of e, for instance, have nothing to do with its decimal expansion (other than the triviality that it never repeats because e is irrational). Mathematicians (and grad students like myself) almost never write something like "e =~ 2.71828...". It's true, but we don't care. There are far more interesting ways to characterize it, such as the base of the unique exponential function which is its own derivative.
Changing from 10 to 16 would not help (or hurt) mathematics in the slightest. Try opening up a serious math book and looking for numerical constants greater than 9 (i.e. ones that would look different in hexadecimal). You won't find very many.
Among the various bases, though, balanced ternary is kind of interesting.
Yo dawg, I heard you like the Ackermann function, so OH GOD OH GOD OH GOD
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.
Building Better Software
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.
There on on 60's in a minute and 60 of those in an hour because we decided there should be.
Actually, it's because the Sumerians and Babylonians used a base-60 counting system.
"...always new atoms but always doing the same dance, remembering what the dance was yesterday." -Richard Feynman
...It's the same argument 'when am I ever going to use algebra/geometry [as taught in high school]'.
As an electrical engineer, in undergrad, we were expected to already know a fairly large amount of algebraic and geometric/trigonometric relationships from high school and we never went over those principles in class. Now, if you're not going into a scientific/engineering/mathematics degree you're probably never going to need to use those principles, but it's a good thing to learn incase you don't know whether you want to be a technical student in college (if you even end up going).
As an electrical-engineering undergraduate ... I would think that most people that go through a pure mathematics degree genuinely enjoy these processes... I can guarantee you that this mental training does give me an edge
As an electrical engineering graduate student I can tell you that I genuinely loath my advanced mathematics courses. I'll say it straight up, they're hard as hell. But I will agree with you that because of those courses I've learned skills that allow me to produce better proofs and quicker understanding of mathematical relations in my linear systems, power systems, and dynamic allocations courses compared to my colleagues who have not taken more rigorous mathematics courses.
I always enjoyed studying with the math students (me being the only non-mathematics graduate student). They always were looking for complete, rationally derived proofs, whereas I would be okay with accepting certain principles without a full proof. I don't think they ever understood how I could just assume certain things were correct and then move on to the next step. That's the difference between mathematicians and engineers; mathematicians want a thorough and rigorous proof and engineers are willing to get "just good enough" on the assumption that someone in the past did their mathematics correctly and their equations are correct.
"Educate the mind but never at the expense of the soul."~Blessed Basil Moreau
I don't know, it just seems obvious.
Um, yeah, when you summarize, you omit details (by definition). When the details you omit are the interesting details, what you're left with is indeed obvious and uninteresting.
It's long been known that primes are more common on the low end. The interesting detail is the mathematical relationship between the actual amounts that end up in the various bins if you divide the range and count them up in each bin. Knowing there will be more in the lower bins doesn't tell you this, it just says there will be more in the lower bins. Benford's law tells you about how many (proportionally) will be in each bin. And it's not in any way an artifict of using a base 10 number system, the relationship remains true regardless of how many bins you use (10, 16, 8, whatever).
"Convictions are more dangerous enemies of truth than lies."
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 (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.
My work here is dung.
Quite the story. More tragic to me is that a man can survive all that, but was done in by unrequited love... I suppose I'd rather be struck by lightning myself.
Actually, if you read the decimals of pi backwards, you'll get all the primes after each other.
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 there's some more data for you. I added that to this spreadsheet.
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:
Here is the list of frequencies next to what my model produced:
I would wager that they are correct. Neat discovery!
My work here is dung.