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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.
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?
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
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.
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
> 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
Where are my mod points when I need them, that's pretty damned interesting.
"linux is just DOS with a UNIX like syntax" -- Galactic Dominator (944134)
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.
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.
...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
The millionth prime 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.
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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.