New Pattern Found In Prime Numbers 509
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.'"
Other bases? (Score:5, Insightful)
When happens with the primes are represented in base-9 or base-11?
Duh (Score:3, Insightful)
Benford's "law" is not a law at all... any exponential distribution will exhibit this behavior.
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.
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.
Re:Why do people study "math" in college? (Score:5, Insightful)
Re:Why do people study "math" in college? (Score:4, Insightful)
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.
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.
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
Re:The real article, and what it does and doesn't (Score:4, Insightful)
> 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)
...and all but one would end with 1 as well.
Re:If you're dealing with phone numbers (Score:3, Insightful)
Where are my mod points when I need them, that's pretty damned interesting.
Re:Other bases? (Score:5, Insightful)
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)
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.
Re:Independent Verification (Score:1, Insightful)
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)
Re:Duh (Score:3, Insightful)
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
Re:Why do people study "math" in college? (Score:3, Insightful)
...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.
Re:Independent Verification (Score:3, Insightful)
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)
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)
Re:Other bases? (Score:3, Insightful)
That is how it seems now isn't it?
However, it has never been proven.
Re:Other bases? (Score:3, Insightful)
Re:Other bases? (Score:2, Insightful)
AND end with 1...this must be a conspiracy
Except for 10, of course.