One page 14 (figure 5) they discover the following fact: the difference between the i'th and (i+1)st primes is about log(i).
That is exactly the prime number theorem I mentioned above, conjectured by Gauss around 1800 and proved in 1896 by Hadamard and de la Valee Pussin.
Writing a paper on the distribution of primes and not referring to that is like writing a paper mentioning a discovery that planets move in elliptical orbits, while being ignorant of Kepler's laws or Newton's explanation of them.
As a number theory graduate student, this looks suspicious. This isn't as bad as last summer, when some string theorists claimed a junk proof of the Riemann Hypothesis, but it's close.
Prime numbers are very hard to tackle. Part of the difficulty in this style of problem, as another post points out, is that they are defined multiplicatively, and yet we here care about additive properties (differences in this case).
I have a few concerns with this paper:
1. They look at a really small number of primes (only 10^7 of them). Many false conjectures have been made that way. The most famous case is with the prime number theorem: it's known that up to x there are about x/log(x) primes, and as x grows this estimate becomes more and more accurate. If you do some tests you'll quickly see that there are more than x/log(x) primes up to x for all x you can test for. This was conjectured to be true for all x, until someone proved that actually the difference (# primes up to x) - x/log(x) changes sign infinitely often. The first change is known to happen before x=10^370 -- but try testing that.
2. They use the ansatz Alog(log(x))+B to fit some function of x (the entropy). But for x in the range of concern (at most 10^8), log(log(x)) is essentially constant. Try graphing that function and you'll see for yourself. For all practical purposes (i.e. unless you can run your computer up to numbers like 10^100), doing curve fitting with this function is very suspicious.
If you want to do a degree programme in parallel with something else, the open university (i.e. courses by correspondence) is the right model. However, this takes time. If you want the degree fast, become a full-time student in a physical university.
You need to consider the following point, which several posts have raised in passing: there is a major difference between a CS degree (which you'd get anywhere else in the world) and a liberal arts degree with a Comp. Sci. major (which you'd get from most US universities). Since you're not in it for the "college experience", but rather for the real education, I'd recommed the former style. If you live outside the US it's not a problem, but otherwise I think some US engineering degrees are more reasonable in this sense -- so look for a university where the CS department belongs to the school of engineering. You can complete that kind of programme in 2 years if you work very hard.
Generally a CS degree is very stressful because it's run like basic training: the university only has 3 years to make programmers out of "civilians". The only way for them to get the programming experience it to give a lot of programming assignments in the courses. If you already know how to write code, you can just breeze though these and concentrate on the theory component (Mathematics, Theory of computation, compilers and the other courses you will take).
If you want to speed up your degree the best way is to spend some time before reading the material. Read a good claculus book (e.g. Courant), Cormen, Licerson & Rivert (Intro. to Algorithms), a good OS textbook, and a Theory book (e.g. Papadimitriou) That will cover most of the core material of a CS degree.
I hope you find some of this useful,
Lior
PS: I know I'm opinionated w.r.t the "liberal arts" concept
I'm from Israel, where I completed a B.Sc. in Mathematics & Physics. Now I'm a graduate student in the Math dept. of an American University.
The difference is staggering. The system I'm used to is similar to what you describe: you get a well-rounded education in your profession (e.g. math or physics) and if you wish you can add a (very) small number of extra courses.
US universities, on the other hand, seem to be driven by the humanities and the concept of the "liberal arts degree". This leads to the dreaded "distribution requirements" which make me inflict calculus on people who'd rather take courses in the humanities, and make science students waste time on humanities courses.
There is quite a marked difference between incoming international and American students in the breadth of knowledge at the beginning of graduate school (and the Americans have to work hard to overcome this gap). In their speciality, the Americans know as much as the international students (more, in many cases) -- they've had the time to take a few advanced courses in one direction. But when it comes to breadth, they just didn't get the opportunity. I took more than 10 courses a semester for 3 years as an undergrad, getting an introduction to almost every branch of math & physics, as well as some advanced courses. You just couldn't do it here (unless you were willing to add even more load).
I'm not saying that there was no freedom in my program of studies, just that the freedom was generally limited to relevant subjects.
In my opinion a "well-rounded" education in the liberal-arts sense belongs in high-school. After that you should be free to pursue whatever studies you wish.
When you go into a new environment, you need an identity. This includes the web. When I shop on the web, I need to use my real-world identity. When I post on/. I can use a/.-generated identity which is less exposed.
What's wrong with a commercial venture that manages identities? You approach this company, and ask them to create you an identity, possibly based on some real-world data like your credit card number. When you interact with a third party you can say "I have personal ID number 57798 issued by that company", together with some documentations (e.g. using public-key certificates). If this third party trusts the company, they will agree that you are who you say you are. This way you can create binding contracts with people you've only met on-line.
Of course, if you couple such a system with a monopoly in some market (e.g. operating systems, mainframes, or insurance) you get in trouble. This is the general problem with monopolies. Also, I'm not sure if I'd use an identity offered by my credit card company since they know enough about me already. If I think some company won't keep my info secret, I won't deal with them, etc. In any case, it's then a matter of consumer choice.
The "let people have IDs on your site" approach doesn't work for sites who who do major business with those people -- you need some third party who'll vouch that these people are genuine.
Remember, the only way to have complete privacy is not to interact with anyone else.
Marybeth Peters, the chief of the United States Copyright Office, said that the exception was still meaningful, even without a market for anti- circumvention devices, because it allowed individuals to figure out for themselves how to go around a technological control measure.
"Many of the people I know can come up with a program to do it themselves, without being in the business of doing it," Ms. Peters said.
So, according to the US copyright office, hacking e-books is a common skill? In fact, a neccessary skill to excersize our rights?
Slashdot Mirror
Another point:
One page 14 (figure 5) they discover the following fact: the difference between the i'th and (i+1)st primes is about log(i).
That is exactly the prime number theorem I mentioned above, conjectured by Gauss around 1800 and proved in 1896 by Hadamard and de la Valee Pussin.
Writing a paper on the distribution of primes and not referring to that is like writing a paper mentioning a discovery that planets move in elliptical orbits, while being ignorant of Kepler's laws or Newton's explanation of them.
Lior
As a number theory graduate student, this looks suspicious. This isn't as bad as last summer, when some string theorists claimed a junk
proof of the Riemann Hypothesis, but it's close.
Prime numbers are very hard to tackle. Part of the difficulty in this style of problem, as another post points out, is that they are defined multiplicatively, and yet we here care about additive properties (differences in this case).
I have a few concerns with this paper:
1. They look at a really small number of primes (only 10^7 of them). Many false conjectures have been made that way. The most famous case is with the prime number theorem: it's known that up to x there are about x/log(x) primes, and as x grows this estimate becomes more and more accurate. If you do some tests you'll quickly see that there are more than x/log(x) primes up to x for all x you can test for. This was conjectured to be true for all x, until someone proved that actually the difference (# primes up to x) - x/log(x) changes sign infinitely often. The first change is known to happen before x=10^370 -- but try testing that.
2. They use the ansatz Alog(log(x))+B to fit some function of x (the entropy). But for x in the range of concern (at most 10^8), log(log(x)) is essentially constant. Try graphing that function and you'll see for yourself. For all practical purposes (i.e. unless you can run your computer up to numbers like 10^100), doing curve fitting with this function is very suspicious.
My take,
Lior
If you want to do a degree programme in parallel with something else, the open university (i.e. courses by correspondence) is the right model. However, this takes time. If you want the degree fast, become a full-time student in a physical university.
You need to consider the following point, which several posts have raised in passing: there is a major difference between a CS degree (which you'd get anywhere else in the world) and a liberal arts degree with a Comp. Sci. major (which you'd get from most US universities). Since you're not in it for the "college experience", but rather for the real education, I'd recommed the former style. If you live outside the US it's not a problem, but otherwise I think some US engineering degrees are more reasonable in this sense -- so look for a university where the CS department belongs to the school of engineering. You can complete that kind of programme in 2 years if you work very hard.
Generally a CS degree is very stressful because it's run like basic training: the university only has 3 years to make programmers out of "civilians". The only way for them to get the programming experience it to give a lot of programming assignments in the courses. If you already know how to write code, you can just breeze though these and concentrate on the theory component (Mathematics, Theory of computation, compilers and the other courses you will take).
If you want to speed up your degree the best way is to spend some time before reading the material. Read a good claculus book (e.g. Courant), Cormen, Licerson & Rivert (Intro. to Algorithms), a good OS textbook, and a Theory book (e.g. Papadimitriou) That will cover most of the core material of a CS degree.
I hope you find some of this useful,
Lior
PS: I know I'm opinionated w.r.t the "liberal arts" concept
Hi,
I'm from Israel, where I completed a B.Sc. in Mathematics & Physics. Now I'm a graduate student in the Math dept. of an American University.
The difference is staggering. The system I'm used to is similar to what you describe: you get a well-rounded education in your profession (e.g. math or physics) and if you wish you can add a (very) small number of extra courses.
US universities, on the other hand, seem to be driven by the humanities and the concept of the "liberal arts degree". This leads to the dreaded "distribution requirements" which make me inflict calculus on people who'd rather take courses in the humanities, and make science students waste time on humanities courses.
There is quite a marked difference between incoming international and American students in the breadth of knowledge at the beginning of graduate school (and the Americans have to work hard to overcome this gap). In their speciality, the Americans know as much as the international students (more, in many cases) -- they've had the time to take a few advanced courses in one direction. But when it comes to breadth, they just didn't get the opportunity. I took more than 10 courses a semester for 3 years as an undergrad, getting an introduction to almost every branch of math & physics, as well as some advanced courses. You just couldn't do it here (unless you were willing to add even more load).
I'm not saying that there was no freedom in my program of studies, just that the freedom was generally limited to relevant subjects.
In my opinion a "well-rounded" education in the liberal-arts sense belongs in high-school. After that you should be free to pursue whatever studies you wish.
Just my rants
When you go into a new environment, you need an identity. This includes the web. When I shop on the web, I need to use my real-world identity. When I post on /. I can use a /.-generated identity which is less exposed.
What's wrong with a commercial venture that manages identities? You approach this company, and ask them to create you an identity, possibly based on some real-world data like your credit card number. When you interact with a third party you can say "I have personal ID number 57798 issued by that company", together with some documentations (e.g. using public-key certificates). If this third party trusts the company, they will agree that you are who you say you are. This way you can create binding contracts with people you've only met on-line.
Of course, if you couple such a system with a monopoly in some market (e.g. operating systems, mainframes, or insurance) you get in trouble. This is the general problem with monopolies. Also, I'm not sure if I'd use an identity offered by my credit card company since they know enough about me already. If I think some company won't keep my info secret, I won't deal with them, etc. In any case, it's then a matter of consumer choice.
The "let people have IDs on your site" approach doesn't work for sites who who do major business with those people -- you need some third party who'll vouch that these people are genuine.
Remember, the only way to have complete privacy is not to interact with anyone else.
Just my rants.
Marybeth Peters, the chief of the United States Copyright Office, said that the exception was still meaningful, even without a market for anti- circumvention devices, because it allowed individuals to figure out for themselves how to go around a technological control measure. "Many of the people I know can come up with a program to do it themselves, without being in the business of doing it," Ms. Peters said.
So, according to the US copyright office, hacking e-books is a common skill? In fact, a neccessary skill to excersize our rights?