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New Pattern Found In Prime Numbers
Posted by
Soulskill
on Sun May 10, 2009 10:34 AM
from the benford-and-sons dept.
from the benford-and-sons dept.
stephen.schaubach writes "Spanish Mathematicians have discovered a new pattern in primes that surprisingly has gone unnoticed until now. 'They found that the distribution of the leading digit in the prime number sequence can be described by a generalization of Benford's law. ... Besides providing insight into the nature of primes, the finding could also have applications in areas such as fraud detection and stock market analysis. ... Benford's law (BL), named after physicist Frank Benford in 1938, describes the distribution of the leading digits of the numbers in a wide variety of data sets and mathematical sequences. Somewhat unexpectedly, the leading digits aren't randomly or uniformly distributed, but instead their distribution is logarithmic. That is, 1 as a first digit appears about 30% of the time, and the following digits appear with lower and lower frequency, with 9 appearing the least often.'"
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Other bases? (Score:5, Insightful)
When happens with the primes are represented in base-9 or base-11?
Re:Other bases? (Score:5, Funny)
It would be bad.
Parent
Re:Other bases? (Score:5, Funny)
Bad as in "cross the streams" bad, or "according to an AC on Slashdot" bad ?
Parent
Re:Other bases? (Score:5, Informative)
Benson's Law is actually independent of the number base used. It wouldn't be much of a mathematical property if it wasn't. No matter how you convert a number, you will always see the same bias.
Parent
Re:Other bases? (Score:5, Insightful)
I don't know; it might be interesting to know that the leading digits of powers-of-k are distributed in some interesting way in base not-k. They obviously all have a leading 1 in base k.
Parent
Re:Other bases? (Score:5, Funny)
I'm pretty sure that in base-2 with no zero-padding, 100% will start with 1. :-p
Parent
Re:Other bases? (Score:5, Insightful)
...and all but one would end with 1 as well.
Parent
Re:Other bases? (Score:5, Funny)
base-9 or base-11?
NEVER FORGET
Parent
Re:Other bases? (Score:5, Informative)
My math degree is getting dusty, but I'm pretty sure that the same pattern could be represented in another base by changing their generalization of Benford's law to include it, and the distribution would look like log(x)/log(9) or log(x)/log(11). Remember, changing the base of a logarithm is easy: for example, log(x)/log(e) = ln(x)
So you get the same distribution, different base.
Parent
Re:Other bases? (Score:5, Informative)
from benfords law link:
"The result holds regardless of the base in which the numbers are expressed, although the exact proportions change."
Parent
Re:Other bases? (Score:5, Funny)
All your base are belong to Benford.
Parent
Stock market analysis? (Score:5, Interesting)
Re:Stock market analysis? (Score:5, Funny)
I am admittedly not a mathematician, but I do have a good understanding of economics and finance, and I am not seeing how a pattern found in prime numbers could have any application to stock market analysis. Where is the interaction between prime numbers and the praxeology of buying and selling securities?
By understanding the patterns in prime numbers you can learn to spot them and avoid the sub-prime mortgage backed securities. Duh.
Parent
Re:Stock market analysis? (Score:5, Funny)
I've always wondering how I could figure out when someone was trying to pass off a list of fraudulent primes. Glad to see that this problem is finally solved!
Parent
Re:Stock market analysis? (Score:5, Interesting)
You're jesting, but I imagine that many fields of encryption would benefit from this, like dual key encryption, where the security lies in the ability to trust that the product really is of two primes, and that factoring this would be extremely time consuming.
Sets with a backdoor inserted may indeed have a different signature, and to be able to quickly see that one set differs would be invaluable. It wouldn't prove anything, but if, say, keys received from a certain company's key generator stood out like a sore thumb in a Benford distribution check, you would have reason to suspect foul play, incompetence or both.
Parent
Cryptography? (Score:5, Funny)
Could this have any applications there?
"Well, I wasn't expecting The Spanish Mathematician . . ."
Re:Cryptography? (Score:5, Funny)
Our two main powers are insight into the nature of primes, fraud detection
and stock market analysis.
I'll come in again...
Parent
If you're dealing with phone numbers (Score:5, Interesting)
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
Independent Verification (Score:5, Interesting)
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.
Re:9999991 (Score:5, Insightful)
Explain one man being hit seven times with lightning. http://en.wikipedia.org/wiki/Roy_Sullivan [wikipedia.org]
Improbable doesn't mean impossible.
Parent
Re:Why do people study "math" in college? (Score:5, Insightful)
Parent
Plenty of reasons people study math (Score:5, Insightful)
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
Parent
Re:9 not too common? (Score:5, Funny)
that makes my /. id even more impressive :)
Parent
On the density of prime numbers (Score:5, Insightful)
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.
Parent
Re:On the density of prime numbers (Score:5, Informative)
Maybe if I had read the prime number theorem, I would have known that it's O(n / log n), which is somewhat bigger...
Parent