10^-20 comes from the typical number of bases used. I forgot the exact number, but we obviously have a trade-off between how certain we want to be of the number's primality and computational time.
That is not true. The number p1*p2*....*pn+1 is either a prime, OR it has a factor that is not one of the p's. In either case, you have a new prime, which as an aside proves that there are infinitely many primes.
There actually are very good algorithms for finding primality. It has reached the point where proving a number prime is MUCH easier than finding any factors of it.
There are two types. One is deterministic, and will give you absolute proof that the tested number is prime. The other type is probability based. These are more popular. The most widely used is known as the Miller-Rabin test. It is known to be absolutely correct for all n 3*10^16. For larger n, it will never report a composite to be prime, but there is a small (around 10^-20) chance the "prime" number will be composite. There are no known prime numbers that Miller-Rabin reports to be composite.
In the case of Mersenne numbers, it's a different story. There is a deterministic algorithm called the Lucas-Lehmer test. This will determine whether 2^p-1 is prime with O-notation p! The catch of course is that it only works for Mersenne numbers.
yes. There is a theorem due to Euclid that every even perfect number (a number which is the product of all of its divisors except itself) is of the form (2^n-1)*2^n. The given form does not apply to odd perfect numbers, but it is unknown whether any odd perfect numbers exist.
The best reason for large Mersenne prime numbers that I can think of is that it gives data for mathematicians to formulate conjectures. There are many consequences to theoretical breakthroughs in the field of prime numbers, especially in the field of encryption, as the RSA algorithm requires large prime numbers.
Note: This new prime number by itself is USELESS for encryption. There are only 42 Mersenne numbers, so they can't be used because there are insufficiently many.
It already kind of phones home. Since the default start page in Firefox is a Mozilla/Google Site, all that Site needs to do is check the User Agent to determine if a patch is necessary. If a user changes their home page manually, presumably they would be smart enough to know how to patch their machine by themselves.
I know that Ted Turner has stepped down. I'm just using him as an example of a very liberal media mogul, to show that network heads don't have to be conservative.
We all know how conservative Ted Turner of CNN is, right?
Why does being owned by a corporation mean that it has to be right wing? Corporations are there to make money. Imagine that you're the head of CNN and you're just handed an exclusive scoop backed by solid evidence that will make Bush look really bad. Imagine the ratings! Even if you like Bush, imagine explaining to other executives (who would sell their mothers for the right price) that you deliberately ignored a source of income because you liked Bush. If your hypothesis was true, why would 60 minutes break the Abu Ghraib prison abuse scandal? Heck, why would CBS use obviously forged documents against Bush in the National Gaurd service kerfluffle?
I agree, Fox News is in general right-wing, more or less so depending on the host. I prefer the Wall Street Journal to any of these networks, though.
I am not exactly a fan of Microsoft, but come on. The knee-jerk anti Microsoft tilt of/. is amazing. I can see it now. There will be some post rated funny talking about how little Gates has spent to vaccinate Windows PCs against viruses, and it will completely trivialize what is being done here. I can't understand how anyone could come up with any negative aspect of this $750 million gift. Does an orphan in Guatemala care about how terrible Windows is? At least give the guy credit where credit is due.
Actually, light does have mass, depending on how you define mass.
There is inertial mass, defined by Newton's second law (F=ma) as the proportionality constant between force and acceleration of an object) Since light doesn't accelerate, this definition doesn't apply.
What does apply is gravitational mass. We know newtons law of gravity, and we can measure how much light is affected by gravity, as in a black hole. This gives us a mass for light, since objects need mass to be affected by gravity. If use this "mass" for the light in other equations to calculate things like the energy of the light (E=mc^2) or the momentum of the light (p=mv), it all works out.
All material objects in the universe have both inertial mass and gravitational mass that are equal within an accuracy of any experiment ever devised. Light is strange in that it only has gravitational mass.
The speed of light can be given in terms of other fundamental electromagnetic constants (1/sqrt(permeability of vacuum * permittivity of vacuum)), but I suspect that this doesn't really answer your question.
Now, the question does have a less profound answer that is not what you have in mind. A meter is DEFINED as the amount of time that light moves in 1/299792458 seconds, so light moves exactly at 299792458 meters per second. The miles per hour speed is just a conversion factor away.
In fact, you can't go faster than light through spacetime because you can't go slower. Even when sitting on your ass reading/., you are moving through time.
This is why time slows as you travel nearer to the speed of light through space, as more of the speed is moving you through the space component of spacetime as opposed to the time component of spacetime.
Look up four-vectors in special relativity for more info.
Agreed. Napster did not reduce piracy when they used a name-based blacklist of files that couldn't be shared.
I'm talking about whatever technology Napster implemented afterwards that eliminated pirated content. Some sophisticated audio analysis technology that I'm too lazy to dig up.
But Grokster is refusing because they will go out of business if piracy is banned from their networks.
what's to prevent a pirate from defrauding SNOCAP into thinking that he represents a label who owns copyright in a given work?
Since the most frequently pirated songs are those of major labels,
I think a phone call would do the trick. There are only so many hit (read: frequently pirated) songs.
Remember, all I know is that Napster did something, and whatever they did, it reduced pirated content on their networks to such a degree that they went out of business. There is something out there that eliminates piracy on one particular network.. MGM of course won't eliminate or even significantly reduce piracy with their tactics, but they can destroy networks trying until they wise up.
I believe that Grokster is centralized. If this is true, than perhaps technology such as SnoCap http://www.businessweek.com/technology/content/dec 2004/tc2004123_8817_tc119.htm
could be used, at least for music files. DRM can work to reduce piracy on a centralized network at least, by requiring downloaders to obtain a license.
Note, these things will certainly drive Grokster out of business like it did for Napster. It will also do little if anything to stem piracy as the file sharers can simply migrate to decentralized and/or encrypted peer-to-peer networks. But I'm just commenting on MGM's demands, not defending their position. MGM wants Grokster out of business. They will do this by making Grokster into Napster, which somehow virtually eliminated copyrighted music in their servers.
The author's summary is correct, but as far as I know, the lawsuit was filed precisely because Grokster refused to incorporate any anti-piracy measures into its network. Grokster will become the next Napster if it does so.
Indeed. St. Kitts and Nevis is not a signatory of any world copyright treaties. This is the reason why part of Sharman networks (which owns Kazaa) is partly located on Nauru (a remote pacific island).
Slashdot Mirror
correction: The form of the number is (2^n-1)*2^(n-1)
10^-20 comes from the typical number of bases used. I forgot the exact number, but we obviously have a trade-off between how certain we want to be of the number's primality and computational time.
GP is right, notice the word OR :)
That is not true. The number p1*p2*....*pn+1 is either a prime, OR it has a factor that is not one of the p's. In either case, you have a new prime, which as an aside proves that there are infinitely many primes.
doh!! My bad.
nope. Try it and see for small n. There are only as many perfect numbers known as there are Mersenne primes known.
There actually are very good algorithms for finding primality. It has reached the point where proving a number prime is MUCH easier than finding any factors of it.
There are two types. One is deterministic, and will give you absolute proof that the tested number is prime. The other type is probability based. These are more popular. The most widely used is known as the Miller-Rabin test. It is known to be absolutely correct for all n 3*10^16. For larger n, it will never report a composite to be prime, but there is a small (around 10^-20) chance the "prime" number will be composite. There are no known prime numbers that Miller-Rabin reports to be composite.
In the case of Mersenne numbers, it's a different story. There is a deterministic algorithm called the Lucas-Lehmer test. This will determine whether 2^p-1 is prime with O-notation p! The catch of course is that it only works for Mersenne numbers.
yes. There is a theorem due to Euclid that every even perfect number (a number which is the product of all of its divisors except itself) is of the form
(2^n-1)*2^n. The given form does not apply to odd perfect numbers, but it is unknown whether any odd perfect numbers exist.
The best reason for large Mersenne prime numbers that I can think of is that it gives data for mathematicians to formulate conjectures. There are many consequences to theoretical breakthroughs in the field of prime numbers, especially in the field of encryption, as the RSA algorithm requires large prime numbers.
Note: This new prime number by itself is USELESS for encryption. There are only 42 Mersenne numbers, so they can't be used because there are insufficiently many.
http://linux.slashdot.org/article.pl?sid=05/02/24/ 1546209&tid=163&tid=147
Gees, not even left the front page yet!
"It's clearly time for the MPAA and RIAA to change their business models and stop suing their customers."
People that don't buy from them are not their customers.
We already have that. It's called Speedstep and it's on Pentium M processors.
It already kind of phones home. Since the default start page in Firefox is a Mozilla/Google Site, all that Site needs to do is check the User Agent to determine if a patch is necessary. If a user changes their home page manually, presumably they would be smart enough to know how to patch their machine by themselves.
I know that Ted Turner has stepped down. I'm just using him as an example of a very liberal media mogul, to show that network heads don't have to be conservative.
We all know how conservative Ted Turner of CNN is, right?
Why does being owned by a corporation mean that it has to be right wing? Corporations are there to make money. Imagine that you're the head of CNN and you're just handed an exclusive scoop backed by solid evidence that will make Bush look really bad. Imagine the ratings! Even if you like Bush, imagine explaining to other executives (who would sell their mothers for the right price) that you deliberately ignored a source of income because you liked Bush. If your hypothesis was true, why would 60 minutes break the Abu Ghraib prison abuse scandal? Heck, why would CBS use obviously forged documents against Bush in the National Gaurd service kerfluffle?
I agree, Fox News is in general right-wing, more or less so depending on the host. I prefer the Wall Street Journal to any of these networks, though.
I am not exactly a fan of Microsoft, but come on. The knee-jerk anti Microsoft tilt of /. is amazing. I can see it now. There will be some post rated funny talking about how little Gates has spent to vaccinate Windows PCs against viruses, and it will completely trivialize what is being done here. I can't understand how anyone could come up with any negative aspect of this $750 million gift. Does an orphan in Guatemala care about how terrible Windows is? At least give the guy credit where credit is due.
Honest mistake on my part. Just a bit tired over here. Ignore parent.
Actually, light does have mass, depending on how you define mass.
There is inertial mass, defined by Newton's second law (F=ma) as the proportionality constant between force and acceleration of an object) Since light doesn't accelerate, this definition doesn't apply.
What does apply is gravitational mass. We know newtons law of gravity, and we can measure how much light is affected by gravity, as in a black hole. This gives us a mass for light, since objects need mass to be affected by gravity. If use this "mass" for the light in other equations to calculate things like the energy of the light (E=mc^2) or the momentum of the light (p=mv), it all works out.
All material objects in the universe have both inertial mass and gravitational mass that are equal within an accuracy of any experiment ever devised. Light is strange in that it only has gravitational mass.
The speed of light can be given in terms of other fundamental electromagnetic constants (1/sqrt(permeability of vacuum * permittivity of vacuum)), but I suspect that this doesn't really answer your question.
Now, the question does have a less profound answer that is not what you have in mind. A meter is DEFINED as the amount of time that light moves in 1/299792458 seconds, so light moves exactly at 299792458 meters per second. The miles per hour speed is just a conversion factor away.
In fact, you can't go faster than light through spacetime because you can't go slower. Even when sitting on your ass reading /., you are moving through time.
This is why time slows as you travel nearer to the speed of light through space, as more of the speed is moving you through the space component of spacetime as opposed to the time component of spacetime.
Look up four-vectors in special relativity for more info.
Agreed. Napster did not reduce piracy when they used a name-based blacklist of files that couldn't be shared.
I'm talking about whatever technology Napster implemented afterwards that eliminated pirated content. Some sophisticated audio analysis technology that I'm too lazy to dig up.
But Grokster is refusing because they will go out of business if piracy is banned from their networks.
what's to prevent a pirate from defrauding SNOCAP into thinking that he represents a label who owns copyright in a given work? Since the most frequently pirated songs are those of major labels, I think a phone call would do the trick. There are only so many hit (read: frequently pirated) songs. Remember, all I know is that Napster did something, and whatever they did, it reduced pirated content on their networks to such a degree that they went out of business. There is something out there that eliminates piracy on one particular network.. MGM of course won't eliminate or even significantly reduce piracy with their tactics, but they can destroy networks trying until they wise up.
I believe that Grokster is centralized. If this is true, than perhaps technology such as SnoCap http://www.businessweek.com/technology/content/dec 2004/tc2004123_8817_tc119.htm
could be used, at least for music files. DRM can work to reduce piracy on a centralized network at least, by requiring downloaders to obtain a license.
Note, these things will certainly drive Grokster out of business like it did for Napster. It will also do little if anything to stem piracy as the file sharers can simply migrate to decentralized and/or encrypted peer-to-peer networks. But I'm just commenting on MGM's demands, not defending their position. MGM wants Grokster out of business. They will do this by making Grokster into Napster, which somehow virtually eliminated copyrighted music in their servers.
The author's summary is correct, but as far as I know, the lawsuit was filed precisely because Grokster refused to incorporate any anti-piracy measures into its network. Grokster will become the next Napster if it does so.
Indeed. St. Kitts and Nevis is not a signatory of any world copyright treaties. This is the reason why part of Sharman networks (which owns Kazaa) is partly located on Nauru (a remote pacific island).