Slashdot Mirror
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.
Explain 9999991 then. :P
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.
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.
Where does Optimus Prime fit in?
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).
http://xkcd.com/435/
That and the fact higher math can be quite beautiful.
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.
"Top * Lists" aren't necessarily unassailable data, but mathematicians, statisticians, and actuaries can do pretty well.
A *lot* of physics would not be possible without mathematics, a lot of chemistry and 'engineering' would not be possible without physics...get it?
Besides it's a beautiful occupation that show truly beautiful things and brings great understanding, it's also very good for learning how to think properly which I really miss in a lot of 'non mathematicians'. (sure they have a lot of knowledge, but they still think in goofed up ways)
This... should not come as a surprise to anyone. We all know primes occur less often as you go higher, which makes sense by the definition of a prime. So there are going to be more primes between 1000 and 2000 than between 2000 and 3000, and so on. Meaning the leading digit will be a 1 more often than a 2, a 2 more often than a 3, etc.
I hope this is more than just the bias you'd expect from the Prime Number Theorem, a corollary of which is that the population of primes decreases exponentially with N.
In each "decade" (10*p.. 10*(p+1)-1, for positive integer p), digits with leading digit 1 come first, and digits with leading digit 9 come last.
For example, with p=2 the numbers (100..199) are in the first group, while (900..999) are in the last group. So you would expect the group with leading digit 1 to have, on average, more primes than the group with leading digit 9.
Rinse and repeat for every decade with that same Prime Number Theorem bias.
Just as in an ideal NFL draft with uniformly competent general managers and no trades, the team picking first in each round is supposed to have a better draft than the team picking last in each round. (Of course, in real life we have the Detroit Lions).
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.?
I call BS, why would primes care what they look like in base10? And how could something like this be equally true for different base notations at the same time, which is what the Wikipedia article claims. I mean does this really apply equally to base 2 binary and base 9?
And a quick glance at the wikipedia list of sequences that are supposed to follow this "Benfords law" shows a lot of areas where the unit of measurement can be chosen somewhat arbitrarily and where humans pick the numbers:
People might irrationally be more likely to say what 100 Kwh? Time to start saving energy. And there might be all sorts of discounts/ tax incentives that kick in at a 100 something per something.
Sounds like house numbers to me, again something entirely under the control of humans and their peculiar preferences for certain base 10 numbers. Though I kind of expect the lack of thirteenth floors to mess up this example.
I am not gonna bother debating whether there a bunches of stock traders who use chicken entrails when deciding when to buy and sell at a time when almost everyone worries that the chicken entail guys might be the more rational ones on the major financial markets.
Of course if humans have to set millions of the prices at which they will trade per day they some of them will go for round numbers and might favor a leading one. We would be loucky if that was the worst thing stock traders did.
Interesting for countries, not for populations for which its easier to decide to invest in and move into and out of like cities. Of course humans if over time millions of people have to make the decision to I migrate into/out of and invest into a town then people will say 95.000 inhabitants? too small to invest/stay, 100.000? thats just big enough.
deaths per what?
Well how many big rivers are there anyway? I will concede this as a coincidence with the note that that might be some unit of measurement preferences going on.
that are not named and might have been cherry picked.
which, if I understand the "Benford distribution" correctly, is somewhat of a circular argument.
I think I will stick to using prime based crypto for now.
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.
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.
How about only people with 5 digit slashdot ids answer this one? Let's skip the chaff and go straight to the wheat.
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....
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.
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
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
Think of it - it has great properties.
- Easily dividable by 2, 3, 4 and 6 ...)
- Matches real relations between time measures well (12 months, 4 seasons, ~360 days,
- Some existing unit systems match it well (think of the relation between day, hour, minute and second)
I use my prime digit to pick my nose...
Some days I get the sinking feeling Orwell was an optimist.
Why do people study "math" in college?
Because they're American?
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.
I find it interesting that the article doesn't prove any theorems.
I don't find that interesting, since the summary and the linked article told me that they didn't prove any theorems. Perhaps you should search for the string "law", since this is an article about Benford's law and not Benford's theorem.
That leaves me thinking: what does this article tell us that we couldn't find out ourselves by ripping through some prime numbers?
That's kind of the point. Anyone can "find out" something for themselves. The hard part is being for the first to do it. Similarly, "what does Thomas Edison's light bulb tell us that we couldn't find out ourselves by playing with some electrical wires and different restiance filaments inside a vacuum?"
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.
Lest you think that there isn't any real work getting done here, I'll go ahead and post part of the article for you:
Significantly, Luque and Lacasa showed in their study that GBL can be explained by the prime number theorem; specifically, the shape of the mean local density of the sequences is responsible for the pattern
They showed that the reason that the primes follow the Genearlized Benford's Law is because of a property of the prime number theorem. The interesting thing here is that you can use their conclusions to quickly see if a set follows the GBL, even for sets that are not prime numbers. In effect, we can now say that we are "100% certain" if a set has special properties, than it follows GBL.
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")
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.
This is slashdot. Some answers are known well before any questions need be asked.
http://home.zonnet.nl/galien8/prime/prime.html
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 surprising this could get attention/published.
And in other news, the second hundred billion digits of pi still could be random.
It's not surprising at all.
Somewhat unexpectedly, the leading digits aren't randomly or uniformly distributed, but instead their distribution is logarithmic.
This is not surprising or unknown. The primes themselves follow a logarithmic distribution (think about probabilities of any number having a divisor [hint: how fast the largest divisor grows, and how fast the number of divisors grow].)
Of course the first digits will also follow a log distribution. The prime numbers in the range 0-9 will have log distribution, the prime numbers in the range 10-99 will have log distribution, then prime numbers in the range 100-999 will have log distribution, etc. The leading digits in each of those ranges will have a log distribution (since the numbers themselves do). Put the ranges together and you get a log distribution for all leading digits.
Not new or unnoticed, and it gives no new results.
This is a non-result. They didn't prove anything. They only did a statistical analysis on a finite number of primes. That primes should obey Benson's law is an obvious conjecture to make and is in fact a strictly weaker statement than the Riemann hypothesis, which most mathematicians already think is probably true.
...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."
Perhaps we shouldn't be applying advance math to matters of finance...
Then why do I still remember BR-549?
...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.
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.
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.
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.
So I grabbed the wrong reply link at the bottom and dumped it into a top level thread you can find here for those still reading this.
Personally, I've been slowing advancing towards a mathematics degree because:
This is not the discovery of a pattern, this is the discovery of the absence of a pattern. They discovered that the distribution of first digits of primes is what it would be in any (scale-invariant) set of random digits (the scale-invariance is why one has to take logs).
Yeah, but she faked it.
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.
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.
I've checked against 50 million primes and it seems to be at 11% I suppose if I had a greater sample it would reach 11.1%
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"
Math is the language of the universe. If you don't want to understand *ANYTHING*, and I mean ***ANYTHING***, about the universe, then don't study math. Otherwise, you pretty much don't have a choice.
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.
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.
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
... and pray tell, what are they?
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...)
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.