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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.'"
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.
It wouldn't change the logarithmic nature of the distribution of the digits, AFAIK.
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.
from benfords law link:
"The result holds regardless of the base in which the numbers are expressed, although the exact proportions change."
Maybe if I had read the prime number theorem, I would have known that it's O(n / log n), which is somewhat bigger...
Numbers are objects, I wish people would understand that numbers are just distinctions. The whole of mathematics is really just a language of form and structure, a system to systematize and decribe structure and forms (relationships are a type of form).
Benford's law works by the observation that, when numbers come up in certain real world contexts, the fluctuations you get in numbers should be proportional to the numbers themselves. Phrased differently, variations tend to be relative, not absolute. Because of this, if you have a very large range of random numbers from many real world measurements, then you would expect the number between t and t*(1.0001) not to vary too much for small changes in t. Let us try to use this observation very coarsely. Among the numbers with 6 digits, the number that look like 1xxxxxx (those between 100000 and 200000) should be about the same the number between 200000 and 400000. The same thing happens with the numbers with 5 digits or 7 digits or n digits (assuming that you have a wide range of random numbers, and the numbers are the kind that come from certain sorts of real world measurements). Additionally, you can get distributions for the first two digits, the first three digits, etc.
This observation doesn't depend on the base that you're working with.
Now, with the prime numbers, they have a distribution that is different from a lot of real world measurement data. The number of primes between n and n+d is approximately d/ln(n), where ln is the log with base e and d is small compared to n. So the number of primes between 500000 and 600000 is about 100000/ln(500000), and the number of primes between 500000 and 600000 is about 100000/ln(600000). By using this, and being slightly more careful, one can determine fairly easily the distribution of the leading terms of the prime numbers.
This is not a hard result. I would say that any professional mathematician who knew about the basic distribution of the primes could derive the distribution of the leading digis of the prime numbers fairly easily if anybody actually asked them to. The reason nobody mentioned this before is that nobody actually cares. While Benford's law does have applications to fraud detection, this new result does not. It's one of those things that makes people say "ooh, a pattern!" but which is just an easy and somewhat mundane corollary to a well known theorem.
While you're absolutely right about the reasoning behind NYC, LA, and Chicago getting 212, 213, and 312, you're a little off on the 989 and 979 area codes, which are much more recent.
In the original system design, all area codes had a middle digit of 0 or 1. The convention was that a middle digit of 1 was used for area codes that only covered part of a state, while a middle digit of 0 was used for area codes that covered entire states. Furthermore, an area code could not begin with a 1 or a 0. and an area code with a middle digit of 1 couldn't have 1 as the third digit. (This left the shortest dial time area code for a statewide code as 201, which went to New Jersey.)
As early as the late 1950s, the idea of single area codes for some states went out the window (with NJ splitting into 201 and 609 in 1958) because of increasing population and proliferation of phone service.
By the late 1980s, the rules were further changed to allow for area codes with middle digits other than 1 or 0. Area codes like 989 and 979 weren't introduced until the late 1980s at the very earliest, by which point very few people were still using rotary phones. At one point, I had heard that the middle digit value of 9 was reserved for the future to allow for four digit area codes, but I can't vouch for the accuracy of that recollection. There are plenty of other rules, some of which you can see summarized here...
-JMP
They are also distributed as Benson's law describes, providing that k is not a rational power of the base. IAAM.
You mean, how many are Mercene primes?
But how many would contain all 1s? Answer that, and provide a proof for your answer, and you'll make math history.
For those who didn't get it: it's not known whether there are infinitely many Mersenne primes, which have this form in binary (they're primes of the form 2^n - 1). Similarly, if you could figure out how many primes have only their first and last bits equal to 1, you would answer a longstanding question about Fermat primes (which are primes of the form 2^n + 1).
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