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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.
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 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
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!
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
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....
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.
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.
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.