The projective line is a one-point compactification of the real line, and afaik there is only one homeomorphism class of simple, connected curves in projective n-space.
Also, it is the Prime Minister who declares war, neither parliment nor the public have to support him. So Tony Blair took Britain into a war based on his own judgment, which he could have done even if a large majority of Britons were opposed to doing so.
There are two possible topologies, or shapes, that are one dimensional
Even allowing for the nonmathematical notation, the statement isn't quite correct.
Such things as figures-of-eight (that is to say, curves that intersect themselves), or shapes that are made up of two curves that never meet (like two circles) are not going to be homeomorphic to either the line or the circle.
If we restrict ourselves to simple (not crossing themselves), connected (in only one piece) curves in real n-space, then we can distinguish two subclasses, the ones like a line, and the ones like a circle.
If it were possible to prove this number were prime, we would have a fantastic way to generate new primes. Take all the primes you know, multiply them and add one. Voila! new prime!
Of course, since the set "all known primes" is not mathematically distinguishable from any subset of primes, it would have to be true that any number which is of the form pqr...z+1 where p,...,z are primes is prime. Of course, this fails, as you can see taking (2)(3)(5)(7)(11) + 1 = 30031 = (59)(509).
Step 4 is incorrect. The fact you are looking for is the fundamental theorem of arithmetic, which says that every positive integer can be factorised into a product of prime numbers (uniquely up to reordering of factors). Plainly the number N would contradict this fact, so it does not exist, so it cannot be constructed, so there are not finitely many primes.
It is incorrect to say that a positive integer that cannot be divided by any prime is prime, since there is only one such number, 1, and it is not prime. Every prime is divisible by itself.
You can say even more. If M can be written as 2^n - 1, then M is said to be a Mersenne number. If M is also prime, then it is a Mersenne prime. For 2^n - 1 to be a Mersenne prime, n must be a prime number, since we have
For instance, 2^6-1 is (in binary) 111111 = 1001 * 111, as predicted by the above factorisation.
If p is a prime number and if 2^p-1 is a Mersenn prime, then, as was pointed out above, 2^(p-1)(2^p-1), is a perfect number. Moreover, if N is an even perfect number, then N can be written (uniquely) as 2^(p-1)(2^p-1) where p is a prime number and 2^p-1 is a Mersenne prime.
Wikipedia has a reasonably intelligible introduction to perfect numbers, and MathWorld contains a proof of why every even perfect number must have the form claimed above.
To see why M = 2^(p-1)(2^p-1) is a perfect number when p, 2^p-1 are primes, it suffices to note that s(n), the function that maps an integer to the sum of its divisors (e.g. s(6) = 1 + 2 + 3+6, s(8) = 1 + 2 + 4+8) is multiplicative in the number-theoretic sense, that is to say s(ab) = s(a)s(b) whenever a, b have no prime factors in common. Then evaluating s(M) is simply a case of evaluating it on the factors, which are relatively prime since one is a power of 2, 2^(p-1), and the other is an odd prime, 2^p-1. s(2^p-1) = 2^p-1 + 1 = 2^p (since we have a prime number), and 2^(p-1) = 2^p -1 is an easy formula that is true of all powers of 2. Hence s(M) = 2^p(2^p-1) = 2 ( 2^(p-1) (2^p-1) = 2s(M). That is to say, the sum of all the divisors of M add up to twice M, and if we leave the divisor M itself out of the sum, we see that M is a perfect number.
I think that in the EU we are mostly fairly happy with the levels of freedom of speech we have. In fact, I believe that the American model is the worse one. I don't think it's tolerable that anyone should be able to preach blind hatred.
When you don't know everything, you know nothing, but you can guess at a great deal.
Everything you said we know about CO2 is true, but until we know everything, there will be space for surprises. I'm not defending the position that we should pump CO2 into the atmosphere here, just saying that all we have today is a fabric of guesswork.
Hello crazy. The US already complies (more-or-less) with what the Security Council recommended. In the US, small arms are not a threat to political stability; in the US the state does regulate the market up to a point (so that people can't own crazy weaponry, but you knew this). Despite the comments of one 'unnamed source' this is a total non-story in the US, as evidenced by the fact that the US supports all the recommendations.
Actually, I am a mathematician of sorts (grad student), but not a number theorist.
If you read my post closely, you'd see that I claimed N! + 2 and N! + 3 were both nonprime (provided N > 2), not that they were prime as you somehow understood.
So the 65536 consecutive numbers 65537! +2 ,..., 65537! + 65537 are all nonprime (the first is divisible by 2, the second by 3, the thrid by 4 etc). Since we also know that there are infinitely many primes, there must be a prime greater than 65537! + 65537. The biggest prime less than 65537! + 2 and the least prime bigger than 65537! + 65537 are consecutive primes which are at least 65536 apart.
Of course, 65537! is a massive number, which is unlikely to crop up in these calculations. There may be other sufficiently large gaps between primes among smaller numbers.
Consider, for instance, the method above applied with 5 in the place of 65536. We see that {6! +2,..., 6! + 5} ={722, 723, 724, 725, 726} are all nonprime, but also 24, 25, 26, 27, 28 are 5 consecutive nonprime numbers.
For where we can expect large 'prime-free' gaps, I'll defer to any number theorist.
Before posting comments, I think you should start by making your comment valid English.
This is a dangling participle, where 'posting' should refer to the the person who can't write in proper English, but the sentence structure associates 'posting' with the subject of the main clause, 'I'. So it means, 'Before I post comments, I think you should...'.
Moreover, the punctuation of 'I noticed that today: your comment...' is bizarre. The colon is seldom used in English, it is generally used to do one of the following: introduce a list; introduce a long quotation; explain the previous sentence. The last use is rare, but still acceptable---one might have:
The delegates didn't spend long in the building after the conference: They all had plans for the weekend.
As a rule, we expect a capital letter to follow a colon.
This isn't quite correct, since the halting problem applies only to a formal system. Life is not a formal system. Consider a real computer posed a real problem. Either the computer succeeds, or the computer falls apart. The problem will not run indefinitely.
Anyway, that waffle aside, my point is that finding errors in engineering is not a formal-system implemented task, but a physical one, so that your argument generally fails. Not that the conclusion isn't true, or anything, but you haven't proved it, you've only given an analogy.
The point here is that we're all better off if kids spend time with their mothers (and fathers); after all, these kids are the ones who'll be paying for your retirement (or at least, paying for the roads when you're retired).
Also, the more stable the family, the less likely the kid is to be a criminal or delinquent.
I'm not trying to exonerate the guy who's going to rob your house, just saying he might have turned out differently.
Also, fuck you when you're old and blind. Why should I subsidise your ass when you never gave me shit?
AFAIK here in Ireland you must have a TV license if you own a device capable of receiving broadcast TV. I haven't contested this in the courts or anything, but that's what I've heard.
The number of people killed by nuclear power rate in the dozens (most at Chernobyl). The number of people killed by coal plants rate in the hundreds of thousands.
But coal plants have been here for much longer, and still provide a large amount of power in developing nations, where safety standards are often not as rigorous as, say, in France.
Also, the number of people killed by Nuclear disasters as a result of cancers which develop only many years later is hard to quantify. (As is, I suppose, deaths due to pollution from coal and oil; but fossil fuel polution has many sources).
The issue here is not whether the motives of the people are honest, or pure. Or whether even the methods they adopt are justified or not. The issue is: 'Is it a threat to the safety of the flight to let this person on'.
And I'm sure anyone who takes a plane would agree with me.
I think Google are behaving responsibly in forestalling the ignorence of others.
These numbers serve only to underestemate the number of non IE browsers in use. Since people (whether warned against it or not) will use Google Zeitgeist for 'research', mostly because it's easy to find and comes from a high-profile source, the result was (and would continue to be) a reinforcement in many people's minds that IE was the 'only' browser.
I'm glad the numbers are gone. I'd like to see a properly-done usage study around here some day.
Slashdot Mirror
The projective line is a one-point compactification of the real line, and afaik there is only one homeomorphism class of simple, connected curves in projective n-space.
Also, it is the Prime Minister who declares war, neither parliment nor the public have to support him. So Tony Blair took Britain into a war based on his own judgment, which he could have done even if a large majority of Britons were opposed to doing so.
There are two possible topologies, or shapes, that are one dimensional
Even allowing for the nonmathematical notation, the statement isn't quite correct.
Such things as figures-of-eight (that is to say, curves that intersect themselves), or shapes that are made up of two curves that never meet (like two circles) are not going to be homeomorphic to either the line or the circle.
If we restrict ourselves to simple (not crossing themselves), connected (in only one piece) curves in real n-space, then we can distinguish two subclasses, the ones like a line, and the ones like a circle.
If it were possible to prove this number were prime, we would have a fantastic way to generate new primes. Take all the primes you know, multiply them and add one. Voila! new prime!
Of course, since the set "all known primes" is not mathematically distinguishable from any subset of primes, it would have to be true that any number which is of the form pqr...z+1 where p,...,z are primes is prime. Of course, this fails, as you can see taking (2)(3)(5)(7)(11) + 1 = 30031 = (59)(509).
Step 4 is incorrect. The fact you are looking for is the fundamental theorem of arithmetic, which says that every positive integer can be factorised into a product of prime numbers (uniquely up to reordering of factors). Plainly the number N would contradict this fact, so it does not exist, so it cannot be constructed, so there are not finitely many primes.
It is incorrect to say that a positive integer that cannot be divided by any prime is prime, since there is only one such number, 1, and it is not prime. Every prime is divisible by itself.
Sorry, that should be '=2M' in the very last equality.
You can say even more. If M can be written as 2^n - 1, then M is said to be a Mersenne number. If M is also prime, then it is a Mersenne prime. For 2^n - 1 to be a Mersenne prime, n must be a prime number, since we have
... + 2^(a))
2^(ab) - 1 = (2^a-1)(2^(a(b-1)) + 2^(a(b-2)) +
For instance, 2^6-1 is (in binary) 111111 = 1001 * 111, as predicted by the above factorisation.
If p is a prime number and if 2^p-1 is a Mersenn prime, then, as was pointed out above, 2^(p-1)(2^p-1), is a perfect number. Moreover, if N is an even perfect number, then N can be written (uniquely) as 2^(p-1)(2^p-1) where p is a prime number and 2^p-1 is a Mersenne prime.
Wikipedia has a reasonably intelligible introduction to perfect numbers, and MathWorld contains a proof of why every even perfect number must have the form claimed above.
To see why M = 2^(p-1)(2^p-1) is a perfect number when p, 2^p-1 are primes, it suffices to note that s(n), the function that maps an integer to the sum of its divisors (e.g. s(6) = 1 + 2 + 3+6, s(8) = 1 + 2 + 4+8) is multiplicative in the number-theoretic sense, that is to say s(ab) = s(a)s(b) whenever a, b have no prime factors in common. Then evaluating s(M) is simply a case of evaluating it on the factors, which are relatively prime since one is a power of 2, 2^(p-1), and the other is an odd prime, 2^p-1. s(2^p-1) = 2^p-1 + 1 = 2^p (since we have a prime number), and 2^(p-1) = 2^p -1 is an easy formula that is true of all powers of 2. Hence s(M) = 2^p(2^p-1) = 2 ( 2^(p-1) (2^p-1) = 2s(M). That is to say, the sum of all the divisors of M add up to twice M, and if we leave the divisor M itself out of the sum, we see that M is a perfect number.
I think that in the EU we are mostly fairly happy with the levels of freedom of speech we have. In fact, I believe that the American model is the worse one. I don't think it's tolerable that anyone should be able to preach blind hatred.
Something is either unique (if there is one of it) or not. Calling something "very unique" is bad style.
My password is quite simple, but this comment is too small to contain it.
When you don't know everything, you know nothing, but you can guess at a great deal.
Everything you said we know about CO2 is true, but until we know everything, there will be space for surprises. I'm not defending the position that we should pump CO2 into the atmosphere here, just saying that all we have today is a fabric of guesswork.
But the theory of the propogation of Tsunamis is, and the problem of warning people is, and GEOLOGISTS ARE NERDS TOO.
Hello crazy. The US already complies (more-or-less) with what the Security Council recommended. In the US, small arms are not a threat to political stability; in the US the state does regulate the market up to a point (so that people can't own crazy weaponry, but you knew this). Despite the comments of one 'unnamed source' this is a total non-story in the US, as evidenced by the fact that the US supports all the recommendations.
Good! Innovation is sure to boom under the science driven administration!
So that's vi over emacs then.
Speaking of which, why can't emacs register votes?
Actually, I am a mathematician of sorts (grad student), but not a number theorist.
If you read my post closely, you'd see that I claimed N! + 2 and N! + 3 were both nonprime (provided N > 2), not that they were prime as you somehow understood.
How long before they place these in the tinfoil?
Soon I won't even be able to trust my hat.
From the article:
... * 65537 + 2 = 2 * (3 * ... * 65537 + 1) ... * 65537 + 3 = 3 * (2* 4 * ... * 65537 + 1)
...
... * 65536 * 65537 + 65537 = 65537(65536!+1)
..., 65537! + 65537 are all nonprime (the first is divisible by 2, the second by 3, the thrid by 4 etc). Since we also know that there are infinitely many primes, there must be a prime greater than 65537! + 65537. The biggest prime less than 65537! + 2 and the least prime bigger than 65537! + 65537 are consecutive primes which are at least 65536 apart.
..., 6! + 5} ={722, 723, 724, 725, 726} are all nonprime, but also 24, 25, 26, 27, 28 are 5 consecutive nonprime numbers.
who can be sure that there aren't two consecutive primes somewhere that are more than 65536 apart
Let's recall elementary number theory:
65537! + 2 = 2 * 3 *
65537! + 3 = 2 * 3 *
65537! + 65537 = 2* 3 *
So the 65536 consecutive numbers 65537! +2 ,
Of course, 65537! is a massive number, which is unlikely to crop up in these calculations. There may be other sufficiently large gaps between primes among smaller numbers.
Consider, for instance, the method above applied with 5 in the place of 65536. We see that {6! +2,
For where we can expect large 'prime-free' gaps, I'll defer to any number theorist.
Before posting comments, I think you should start by making your comment valid English.
This is a dangling participle, where 'posting' should refer to the the person who can't write in proper English, but the sentence structure associates 'posting' with the subject of the main clause, 'I'. So it means, 'Before I post comments, I think you should...'.
Moreover, the punctuation of 'I noticed that today: your comment...' is bizarre. The colon is seldom used in English, it is generally used to do one of the following: introduce a list; introduce a long quotation; explain the previous sentence. The last use is rare, but still acceptable---one might have:
The delegates didn't spend long in the building after the conference: They all had plans for the weekend.
As a rule, we expect a capital letter to follow a colon.
This isn't quite correct, since the halting problem applies only to a formal system. Life is not a formal system. Consider a real computer posed a real problem. Either the computer succeeds, or the computer falls apart. The problem will not run indefinitely.
Anyway, that waffle aside, my point is that finding errors in engineering is not a formal-system implemented task, but a physical one, so that your argument generally fails. Not that the conclusion isn't true, or anything, but you haven't proved it, you've only given an analogy.
The point here is that we're all better off if kids spend time with their mothers (and fathers); after all, these kids are the ones who'll be paying for your retirement (or at least, paying for the roads when you're retired).
Also, the more stable the family, the less likely the kid is to be a criminal or delinquent.
I'm not trying to exonerate the guy who's going to rob your house, just saying he might have turned out differently.
Also, fuck you when you're old and blind. Why should I subsidise your ass when you never gave me shit?
AFAIK here in Ireland you must have a TV license if you own a device capable of receiving broadcast TV. I haven't contested this in the courts or anything, but that's what I've heard.
The number of people killed by nuclear power rate in the dozens (most at Chernobyl). The number of people killed by coal plants rate in the hundreds of thousands.
But coal plants have been here for much longer, and still provide a large amount of power in developing nations, where safety standards are often not as rigorous as, say, in France.
Also, the number of people killed by Nuclear disasters as a result of cancers which develop only many years later is hard to quantify. (As is, I suppose, deaths due to pollution from coal and oil; but fossil fuel polution has many sources).
The issue here is not whether the motives of the people are honest, or pure. Or whether even the methods they adopt are justified or not. The issue is: 'Is it a threat to the safety of the flight to let this person on'.
And I'm sure anyone who takes a plane would agree with me.
I think Google are behaving responsibly in forestalling the ignorence of others.
These numbers serve only to underestemate the number of non IE browsers in use. Since people (whether warned against it or not) will use Google Zeitgeist for 'research', mostly because it's easy to find and comes from a high-profile source, the result was (and would continue to be) a reinforcement in many people's minds that IE was the 'only' browser.
I'm glad the numbers are gone. I'd like to see a properly-done usage study around here some day.