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Math News

New Pattern Found In Prime Numbers 509

Posted by Soulskill
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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New Pattern Found In Prime Numbers

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  • Other bases? (Score:5, Insightful)

    by wiredlogic (135348) on Sunday May 10, 2009 @09:38AM (#27896535)

    When happens with the primes are represented in base-9 or base-11?

  • Duh (Score:3, Insightful)

    by Anonymous Coward on Sunday May 10, 2009 @09:41AM (#27896567)

    Benford's "law" is not a law at all... any exponential distribution will exhibit this behavior.

  • Re:9999991 (Score:5, Insightful)

    by Aranykai (1053846) <slgonser@gmailERDOS.com minus math_god> on Sunday May 10, 2009 @09:43AM (#27896583)

    Explain one man being hit seven times with lightning. http://en.wikipedia.org/wiki/Roy_Sullivan [wikipedia.org]
    Improbable doesn't mean impossible.

  • Re:Other bases? (Score:5, Insightful)

    by jonaskoelker (922170) <jonaskoelker&gnu,org> on Sunday May 10, 2009 @09:54AM (#27896689) Homepage

    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.

  • by wjh31 (1372867) on Sunday May 10, 2009 @09:56AM (#27896711) Homepage
    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
  • by MikeUW (999162) on Sunday May 10, 2009 @09:58AM (#27896729)

    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.

  • by jonaskoelker (922170) <jonaskoelker&gnu,org> on Sunday May 10, 2009 @10:05AM (#27896779) Homepage

    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.

  • by sirwired (27582) on Sunday May 10, 2009 @10:07AM (#27896785)

    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

  • by quarrel (194077) on Sunday May 10, 2009 @10:10AM (#27896821)

    > 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

  • Re:Other bases? (Score:5, Insightful)

    by dynamo52 (890601) on Sunday May 10, 2009 @10:39AM (#27897003)

    I'm pretty sure that in base-2 with no zero-padding, 100% will start with 1.

    ...and all but one would end with 1 as well.

  • by Sir_Lewk (967686) <sirlewk.gmail@com> on Sunday May 10, 2009 @10:46AM (#27897061)

    Where are my mod points when I need them, that's pretty damned interesting.

  • Re:Other bases? (Score:5, Insightful)

    by Sparr0 (451780) <sparr0@gmail.com> on Sunday May 10, 2009 @11:14AM (#27897305) Homepage Journal

    The faux concern, or misplaced real concern, so many people show over 9/11 has made it a relevant target for such jokes since 9/12.

  • Re:Other bases? (Score:2, Insightful)

    by Anonymous Coward on Sunday May 10, 2009 @11:20AM (#27897365)

    It was long enough by about a week later. This is the internet, on the internet anything more than a month old is ancient history.

  • by Anonymous Coward on Sunday May 10, 2009 @11:31AM (#27897451)

    The ratio will have a huge amount to do with where you stop. Stop with a prime starting with 1, like you did, and the 1 "probability" will be very high. Stop with a prime starting with 2 and things will be different.

    I find it very hard to believe these ratios actually converge independent of where you stop, which would make TFA BS. Infinite probability distributions over the natural numbers usually don't converge.

  • Complete bullshit (Score:5, Insightful)

    by gnasher719 (869701) on Sunday May 10, 2009 @12:04PM (#27897663)
    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.
  • Re:Duh (Score:3, Insightful)

    by jstott (212041) on Sunday May 10, 2009 @12:13PM (#27897737)

    Benford's "law" is not a law at all... any exponential distribution will exhibit this behavior.

    A law, as the word is commonly used in math and physics, is a mathematical expression of a universal relationship. As you say, Benson's law is a property of any exponential distribution, so we agree it's universal. Why then can't we call it a law? Just because it's obvious after you understand it doesn't make it any less a law.

    -JS

  • by WCguru42 (1268530) on Sunday May 10, 2009 @01:01PM (#27898167)

    ...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.

  • by Halo1 (136547) <jonas.maebe@NospAm.elis.ugent.be> on Sunday May 10, 2009 @01:05PM (#27898219) Homepage

    The millionth prime [utm.edu] is 15,485,863. This means that he considered ~5.5 million more numbers that start with a 1 (10 million - 15.5 million) than numbers that start with any other digit.

  • Re:Duh (Score:1, Insightful)

    by Anonymous Coward on Sunday May 10, 2009 @02:06PM (#27898715)

    As you say, Benson's law is a property of any exponential distribution, so we agree it's universal. Why then can't we call it a law?

    What is Benson's law, is that a law saying that any exponential distribution is distributed exponentially?

    Benford's Law says that "in lists of numbers from many real-life sources of data, the leading digit is distributed in a specific, non-uniform way". That word "many" means it's not universal, so it's not a law.

  • Re:9999991 (Score:3, Insightful)

    by BaldingByMicrosoft (585534) on Sunday May 10, 2009 @03:54PM (#27899425)
    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.
  • Re:Other bases? (Score:3, Insightful)

    by nog_lorp (896553) on Sunday May 10, 2009 @06:02PM (#27900289)

    That is how it seems now isn't it?

    However, it has never been proven.

  • Re:Other bases? (Score:3, Insightful)

    by DavidShor (928926) <supergeek717.gmail@com> on Monday May 11, 2009 @01:53AM (#27903143) Homepage
    You should see Category Theory, "Well, a morphism is when you've got some things, and then you end up with some stuff...."
  • Re:Other bases? (Score:2, Insightful)

    by jonnat (1168035) on Monday May 11, 2009 @07:52AM (#27905073)

    AND end with 1...this must be a conspiracy

    Except for 10, of course.

The trouble with opportunity is that it always comes disguised as hard work. -- Herbert V. Prochnow

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