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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.
with 9 appearing the least often
Maybe they didn't count high enough? I wouldn't blame them, I get tired of computing primes by 7...
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
When she was done I wondered if she was going to be unemployed in a few years, because I could not see how her knowledge had any application in the real world.
Your anecdote does not contain any interesting information. What was her "esoteric math" about?
"You're right," Fisheye says. "I should have set it on 'whip' or 'chop.'"
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!
TFA does provide many details, but I believe everyone doing maths is taught that the number of primes before some number n is approached by n/ln n ( http://en.wikipedia.org/wiki/Prime_counting_function). This has been known for more that a century.
As specified by this formula, the prime density decreases, so it seems obvious that if one considers all primes with a set number of digits, fewer start with a nine than with a one.
Maybe I'm missing something but this does not seem to provide any new information or new pattern.
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.
"Top * Lists" aren't necessarily unassailable data, but mathematicians, statisticians, and actuaries can do pretty well.
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
Prime numbers follow a logarithmic curve!!
Film at 11.
Old?
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
Yeah, it only looks like that because started finding primes from 1 up. If we started finding them from infinity down...
Fuck systemd. Fuck Redhat. Fuck Soylent, too. Wait, scratch the last one.
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.
Harsh critic--the article doesn't boast about solving theorems or offering 100% certain proofs. It is sufficient to bring to greater notice that this pattern, which had gone unnoticed, has now been noticed. Maybe someone will do something further with it. Sooner or later it's likely that this piece of information will get incorporated into something economically useful. But for now, as pure science, noticing the pattern that had not been noticed before is good enough for publication.
Could this be the beginnings of a non-quantum solution to, say, the problem of factoring large numbers? Not a solution itself, but the beginnings of a method for breaking RSA without resorting to the use of q-bits et. al.?
Explain one man being hit seven times with lightning.
Easy. He lied about the last six strikes.
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.
On the other hand, in binary it will be 1 in all cases. Time for a new law? I don't think so...
Most ACs are not even worth the keystrokes to insult them. Be generically insulted by this and ignored otherwise.
Isn't this just a consequence of prime numbers getting sparser as you climb higher?
i.e, there are 135 primes between 1000 and 2000, and there are 127 between 2000 and 3000.
http://www.theregister.co.uk/2009/03/20/top_model_stampede/page3.html
C-x C-s C-x k
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.
It's already obvious that as you increase in size, the primes are further apart (because there are more and more lower numbers that are potential factors).
So given any range of numbers from (10^x) to (10^(x+1)), wouldn't you expect that the density of primes in the bottom end of that range would be higher than at the top end?
I don't know, it just seems obvious. It's an artifact of using a base 10 number system.
Now if you used binary, you wouldn't see this effect. That is, a range from (2^x) to (2^(x+1)) all "start" with the digit 1. Also, if you used a base-infinity number system (where every number has it's own symbol then you also wouldn't see this effect because the maximum number of possible primes in any given "leading digit category" is effectively 1.
"I have never let my schooling interfere with my education." - Mark Twain
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.
I bet this law doesn't apply in base 2 ...
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....
According to National Geographic's Flash Facts About Lightning, the odds of being struck in a lifetime is three-thousand to one.
That sounds surprisingly likely... but I'm inclined to think he was lying.
I mean, doesn't lightning heat the surrounding air enough to melt sand into glass? We're talking thousands of degrees here. How could someone survive that ONCE, let alone 6 more times?
Fact: Everything I say is fiction.
The set of integers is countable, and any subset of it, including the set of primes beginning with a particular digit, is also countable. One can always define a mapping of any infinite subset of any countable set to the complete set of positive integers, which means, perhaps rather counterintuitively, that they actually have the exact same number of elements.
File under 'M' for 'Manic ranting'
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
Hey, I wonder if this holds true beyond Skewes number? (I don't remember all the particulars, but seemingly primes start to become more common beyond some incredibly high value. I vaguely remember reading an article by Isaac Asimov in SF monthly on the subject. At that time, Mr. Asimov suggested that there would be another point where that would reverse, but (with his dry sense of humor) he suggested that attempting to calculate Skewes number already left him skewered, and computing that second reversal point left him super-skewered.
Since numbers themselves don't mean anything at all. They are just abstract tokens we manipulate in a standardized manner in order to try to understand the world around us. Just like a bit doesn't care if it's in a silicon ram chip, flash memory, a "hole" in a reflective surface on a DVD, a vacuum tube or a bead on an abacus. In itself it's meaningless.
To assume that the distribution of numbers means anything at all outside the fact that we humans measure things in "units", start counting from zero and the first digit is the number "one", and we usually try to use a unit that is close to the threshold of detection for our equipment is contemplation of the navel to the, well, first degree.
Seven puppies were harmed during the making of this post.
That's an anomaly. Sullivan clearly refused Zeus' advances, and we all know how vindictive that bastard can be.
Trust me, kids; don't drink and post.
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.
Others have replied to this question well enough. It's the same argument 'when am I ever going to use algebra/geometry [as taught in high school]'. First colleges aren't trade-schools. Their job is to give you a broad foundation, and more importantly, they teach you how to think.
Math teaching you profoundly how to think. As an electrical-engineering undergraduate that doesn't directly use any of his main teachings, I still revisit all my text-books as one might revisit a game of sudoku. Being able to 'prove' a relationship, or walk through a design process requires a great deal of concentration and mental flexing.
I would think that most people that go through a pure mathematics degree genuinely enjoy these processes (at least I do).
I can guarantee you that this mental training does give me an edge over high-school, or even non-mathematically rigorous colleges in my field. A complex business process is met by my colleges with 'that gives me a headache', or 'lets take a break'. Which frustrates me, who would rather dig in deep and long.
Believe me, it isn't because I'm any smarter than them. I strongly believe it's because I had a background in complex problem solving - given by my high school and there-after my university.
-Michael
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.
My take is that he probably lied about the last five and gave it up when the media stopped paying attention to him.
It doesn't heat the air enough, but it will melt sand where the bolt itself travels through it. A human is also a much better conductor than sand, which IIRC means theres going to be less heat produced.
If sharing a song makes you a pirate, what do I have to share to be a ninja?
Primes by definition are numbers not covered in a repeating sequence. So to identity them in number line:
Make N the last prime, remove every Nth whole number, then the first remaining number is the next prime. continue.
[example]
remove every 2 entry then the first remaining number, and next prime is 3.
remove every 3 entry then the first remaining number, and next prime is 5.
remove every 5 entry then the first remaining number, and next prime is 7.
remove every 7 entry then the first remaining number, and next prime is 11.
remove every 11 entry then the first remaining number, and next prime is 13.
I have theorized (just a simple man) that primes could be identified by:
...an optimus discovery.
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
Don't mind them, they're from Barcelona
Never underestimate the bandwidth of a station wagon full of tapes hurtling down the highway. -- Andrew S. Tanenbaum
I use my prime digit to pick my nose...
Some days I get the sinking feeling Orwell was an optimist.
This same phenomenon applies to the physics demonstrations by which a man's arm can be covered in water then dipped in molten lead, or a droplet of liquid nitrogen can roll around in the palm of your hand, both completely safe if performed carefully. Massive temperature gradients can exist in very small spaces, and as long as you are on the safe side of that gradient, no matter how narrow it is, you will be OK. Sure, the air might be a million degrees 1mm from your skin, but if the air touching your skin is 100 degrees then you will just feel slightly warm.
Practically all of physics even. I tend to jokingly refer to physics as applied math.
I can't tell whether the original poster of this thread is trolling or is just incredibly stupid (or both, which might be most likely, insomuch as trolling is stupid), but I'd say that in general math is a superb undergraduate major for quiet practical reasons (independent of the abstract beauty).
People entering undergrad school are often still a bit hazy on exact career goals. In that case, math can be an excellent choice because it is good base preparation for many, many fields - darn near anything scientific or technical, and some other areas as well. I often advise people who are still waffling about the details that math is a good default for undergrad, moving into something more specialized either in grad school or later in their undergrad work.
primes are beleived to be distributed as
#primes x ~ x/ln(x)
so the number from 0 to 10 is something like
~4
10 to 100:
~17
100 to 1000:
~123
1000 to 10000:
~940
as can be seen the number is more linear than logarithmic.
Bensons law arises when things are distributed logarithically.
thus the appearance of bensons law seems like it may be a suprise
Some drink at the fountain of knowledge. Others just gargle.
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
The number of primes pi(x) not exceeding x is approximately x/ln(x). (Source: Graham, Knuth and Patashnik, "Concrete Mathematics")
Besides, you can easily count to 12 -- you have 12 phalanx on your hand if you count them with your thumb (three on each finger thumb excluded). That way you can count up to 144 if you use both hands wisely.
Ok but, degrees, minutes, seconds or days minutes seconds are also kinda arbitrary. There on on 60's in a minute and 60 of those in an hour because we decided there should be.
It would be more natural to measure these things in radians or some other multiple of Pi. 2(Pi) would come up quite a bit so being divisible by 2 is a nice property but both base 2,8,10 and base 16 would also give you that will retaining the advantages they offer.
Repeal the 17th Amendment TODAY! Also Please Read http://www.gnu.org/philosophy/right-to-read.html
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
Given enough time, improbable becomes eventual.
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.
What implications does this have on crypto? Since a lot of decent crypto is based on random number generation will this render modern cryptographic techniques useless? Will there be some emperical way to determine the remaining digits of a random number based on Benfords posit?
It's not that arbitrary. The fundamental reason for that choice was to make it easily divisible. 12 is evenly divisible in a number of ways, but not by five. But multiply 12 by 5 and you get 60, which is evenly divisible by everything 12 is, and by five as well.
Yes, there are 60 minutes and seconds because we decided there should be, but it was by no means an arbitrary decision. There were good, solid mathematical reasons why those numbers were chosen.
"Convictions are more dangerous enemies of truth than lies."
"One in a million chances crop up nine times out of ten."
"Convictions are more dangerous enemies of truth than lies."
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
And the entry for Benson's law?
Confucius say, "Find worm in apple - bad. Find half a worm - worse."
>Ok but, degrees, minutes, seconds or days minutes seconds are also kinda arbitrary.
More like an artifact of human proportions, than merely "arbitrary".
There is a case to be made for a natural tendency toward a twelve month calendar or a twenty-four hour day, even when people didn't represent such things.
-fb Everything not expressly forbidden is now mandatory.
I'm trying to work out what body parts a normal human has 12 or 24 of. Certainly none that are external and easily visible. Also the units appear to be measures of time. I'm not sure what the human body's clock speed is, but whatever you choose it's probably quite variable and we aren't consciously aware of it.
Confucius say, "Find worm in apple - bad. Find half a worm - worse."
I call BS, why would primes care what they look like in base10?
They don't.
And how could something like this be equally true for different base notations at the same time, which is what the Wikipedia article claims.
Quite easily, when you're talking about first digit distributions (and we are), which is basically saying lower magnitude numbers are more frequent than higher magnitude ones. This would (to most people, not you obviously but to most other people) be obviously not influenced by what base you choose to use in expressing the numbers. If lower magnitude numbers are more common than higher magnitude ones, this is simply true regardless of base, so no matter what base you pick, lower magnitude first digits will be more common, and indeed in this case, the distribution of their commonality will vary logarithmically across the digits, regardless of how many of them there are in the base you have chosen, because that's a feature of the set of numbers, not an artifact of their expression.
I mean does this really apply equally to base 2 binary and base 9?
Yes, although base 2 is the degenerate case (all prime numbers start with 1, which is trivial but does satisfy the law).
which, if I understand the "Benford distribution" correctly, is somewhat of a circular argument.
Sort of. To the extent that in any analytic theorem, the truth of the conclusion is essentially contained in the premises, you're just teasing out and making plain what was already there. This is true of all mathematical theorems. That's what makes it analytic rather than synthetic (empirical).
I think I will stick to using prime based crypto for now.
Of course, why wouldn't you? Nothing in the article has any implications for cryptography.
"Convictions are more dangerous enemies of truth than lies."
...to me, Benford's law always seemed completely obvious. I didn't think that someone would give it a name and publish it as a great new concept.
I mean, how can you not immediately notice that, when you try to count to 1000, then up to a million, and play a bit with numbers, as a child?
Ok, someone had to notice that it is a concept, and be the one who writes it down. I give credit for that. But the rest...
Any sufficiently advanced intelligence is indistinguishable from stupidity.
Having a lower /. ID does not in any way signify expertise.
Besides, those 5 digit people are n00bs... :p
"Convictions are more dangerous enemies of truth than lies."
Searching on prime numbers and Benford's law, the first page includes a hit on the book
Prime Numbers by David G. Wells which on page 17 talks about Benford's law and the distribution of first digits of primes.
That way you can count up to 144 if you use both hands wisely.
Pfft. Binary allows up to 1023 with two hands.
What, you're not dextrous enough to use three positions per finger and count to 59,048?
I find that more than three positions per finger becomes difficult to maintain reliably.
I think your sample is badly biased by ending on the 1,000,000th prime, since essentially all your primes are then those between 10,000,000 and 15,485,863 which is the last prime in the first million file.
It took me a while to notice this, too, since any sampling which doesn't sample exact decades is badly biased. By the time you get all the primes less than 1e9, the distribution is very flat. Here are the stats:
first digit histogram [6003531, 5837665, 5735086, 5661135, 5602768, 5556434, 5516130, 5481646, 5453140]
fractions:
0.118069263338
0.114807236968
0.112789853038
0.111335485585
0.110187602998
0.109276369051
0.108483724924
0.107805540623
0.107244923476
I am running through all the primes less that 1e11 right now, and will post that later.
Pick a random spot on a slide-rule. You are more likely to hit a region where the mantissa starts with 1 than any other region. And as posters have already pointed out, a base-2 slide-rule would guarantee that the first digit is 1. There is nothing magic about Benford's law, it only shows that random numbers based on measurements are logarithmically distributed.
Who asked for your opinion? :-)
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.
Personally, I've been slowing advancing towards a mathematics degree because:
Well, your fingers plus your hands makes twelve. But perhaps you'd be better off with a sign flag or something...
"You're right," Fisheye says. "I should have set it on 'whip' or 'chop.'"
That two sets A and B are both countably infinite doesn't necessarily mean that the probability of a random element from B being in A is undeterminable. To take a simple counterexample: the even integers are a countable subset of the integers, and the probability that a randomly selected integer is even is 50%.
The real reason not to study "math" in college in the US is because of immigration laws giving people with doctorates in mathematics preferred status for immigration.
That means the supply of people with those degrees is often far higher than the demand, as one portion of the population is assessing their choice of degree taking into account a secondary positive effect.
There were apparently psychological experiments in the 90s, where people would be deprived of any time cues (sunlight, clocks, television, radio, etc). Their body clock, IIRC, moved to 26 hours for a full wake/sleep cycle.
It's a small world and it smells funny; I'd buy another if it wasn't for the money; Take back what I paid (SoM)
Most people who believe that they were struck by lightning were actually only near a lightning strike and got zapped by induction or step potential. The chances of surviving an actual strike are very poor.
Warning: this article may contain humor, sarcasm, parody, and perhaps even irony. Read at your own risk.
Works reading forward, too. You just have to find the right place to start...
Warning: this article may contain humor, sarcasm, parody, and perhaps even irony. Read at your own risk.
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.
This is the funniest thing I've ever read on ./ - mods are all asleep at the wheel.
What fraction of all positive integers (represented in base 10) start with a "1", or is the value undefined because the limit fails to converge? In the latter case, is there some meaningful average of the upper and lower bounds?
you guys dont have enough to do.
This seemed like a reasonable sig at the time.
Decoded, the pattern spells out the message: "Haha, it's me, God, I existed along"
Four you.
"Screw Sun, cross-platform will never work. Let's move on and steal the Java language." - Visual J++ Product Manager
Obeying Benfords law isn't a pattern - it's the absence of a pattern. If the sequence of leading digits of prime numbers didn't follow Benford's Law, that would be a pattern, and a very interesting and peculiar one - there would have to be some underlying reason for such non-randomness.
From experience, when unhindered by the outside world, I run at around 25 - 25.5 hours per wake/sleep cycle.
So much for public key cryptography!
Isn't this property a simple corollary of one of the standard algorithms used for finding primes?
I.e. 2 is a prime, strike 4, 6, 8, and all other powers of two.
3 is a prime, strike 6, 9, 12 and all other powers of three.
And so on.
Each of the steps will remove numbers from the list at a fixed interval. Meaning the the distance between primes increase as the numbers grow larger. A repeating pattern based on the previous prime numbers.
Since the distance between the numbers grow, you will have more numbers in the beginning of the interval.
I lost my sig.
From quickly glanced through the paper, it doesn't seem to me that the authors have actually proved any theorems - at least not in the mathematical sense of the word "proof." They *have* provided ample numerical evidence that indicates Benford's Law applies to primes, but their explanation for why this is so is because the primes have a 1/log(x) density - nothing deeper than that. It's a cool paper to be sure, but there doesn't seem to be much significance for mathematics.
Why do people study "math" in college?
This is the wrong place to be asking questions like that. You'd get better answers at your local sports bar. Oh, by the way, don't ask people who climb mountains why they studied mountain climbing in school.
It has long been known that the frequency of primes decrease as values increase. That is, there are likely to be more primes between 1 and 101 than between 1000001 and 1000101. It simply makes sense then that for any number of digits, the number of primes between 1x10^n and 2x10^n would be greater than the number of primes between 9x10^n and 10x10^n. Given that we have formulas for predicting this decrease in frequency, finding the actual distribution should be simple. Was there something unexpected about the actual results that were found?
I often don't like the choices people make, but I like the fact that people make choices. That's why I'm a conservative.
You people still sleep?
A property which can be applied to log-normal-spaced distributions of numbers has been applied to a log-normal-spaced distribution of numbers.
News at 11.
-- I was raised on the command line, bitch
Yep, the trick is to be sure it is exactly a once in a million chance (borrowed shamelessly from Terry Pratchett, of course).
"How about only people with 5 digit slashdot ids answer this one? Let's skip the chaff and go straight to the wheat."
No, I vote to have this answered by just the ones with prime Slashdot ID's.
If I had to a-priori guess how the first digit of prime numbers were distributed then I would guess it would be some version on Zipf's law. Which indeed proves to be the case. What would have been more interesting if it failed to follow the rule which would have perhaps given some more insight into the distribution. Or maybe I've missed something.
So its sort of an anti-result - every thing is as predicted. In other news: the sun will rise tomorrow.
There are four sorts of people in the world: fools, lunatics, idiots and morons. - Umberto Eco, Foucaut's pendulum.
I'll bet you wish /. had one right about now, or did you miss the part about 10 ?
Good judgement comes from experience, and experience comes from bad judgement.
- W. Wriston, former Citibank CEO
On the contrary, a set can have measure 0 and be non-empty...(What is the probability of selecting a prime number over the set of integers? Surely there exist primes...)
I tend to run 28 when given the opportunity. Granted, I can't get those around me to let me run that schedule, and I am still running the Uni gamut, so I'm not ready to try for freelance work just yet...
2^3 * 31 * 647
Add sufficient caffeine tablets to their coffee. They will soon switch to 28 hour days.
It's a small world and it smells funny; I'd buy another if it wasn't for the money; Take back what I paid (SoM)
I was thinking more of a lunar calendar than of body parts or the "human's body clock speed"
It's pretty hard to make a case that there's anything but 4 seasons and 12 lunar cycles throughout them.
So there's a natural perception of the synodic calendar.
When your civilization invents the Sundial, the angle of 15 degrees derives naturally from it.
Measurement systems based on the numbers 60 tend to be quite easy to construct. You don't need superstition and numerology to come up with a rationale for sexagesimal subdivision. It's probably no more mysterious than "12 is easily divided into various segments."
I suspect that early civilizations were far more interesting in divining *direction* than *time*. If you know the time of day within 4 hours, and the season within a few weeks, you'll be fine. But knowing what direction to walk (or sail!) is a much bigger deal. And when you're working that out with sticks and strings, lots and lots of things divide by 60 and factors of 60.
I didn't make clear what I meant by "human proportions."
-fb Everything not expressly forbidden is now mandatory.
It is still difficult for me to imagine something as energetic as a lightning bolt not burning a human being to a crisp on the basis that they are reasonably conductive... But very interesting. Thank you for the gem.
Fact: Everything I say is fiction.
Too old. You have triggered some old and long unused neurons. I have not thought of length, area, volume... "measures" of sets in more than 25 years.
I still like the post you originally replied to; wish I had some mod points. That was funny.
Good judgement comes from experience, and experience comes from bad judgement.
- W. Wriston, former Citibank CEO
Sorry guys, but it would actually be surprising if they didn't follow Benford's law. This is a non-issue as far as mathematics goes.
Those were the days! Finishing my dissertation I ended up in a cycle something like work 16, sleep 10, wake 6, sleep 6.
Confucius say, "Find worm in apple - bad. Find half a worm - worse."