No, it is not equally dodgy. The fundamental theorem of arithmetic states that a composite number may be written (uniquely, apart from rearranging the factors) as a product of primes. Hence, the given number has a prime factor not in the list, contradicting the assumption.
I prefer a slightly different way of looking at the proof. Claim: The list of primes is infinite. This is equivalent to claiming that for any prime p, there exists another prime p' > p. Proof: Let q be the successor of the product of all primes up through and including p. Then, if q is prime, clearly q > p. If q is not prime, then by the fundamental theorem of arithmetic, it has a prime factor p'. But, p' cannot be any of the prime factors through p. Therefore, p' > q. Q.E.D.
In this way, we don't have a proof by contradiction.
That is true. Nevertheless, you made the additional claim that a certain number was prime, and used that FALSE claim to arrive at your contradiction. Your proof was incomplete, so I corrected it.
First, I assume you mean twin primes. Proving infinite primes is trivial and from ancient Greece. It is a proved fact that there are arbitrarily large gaps in the prime sequence (i.e. infinitely large gaps). And that primes get rarer and rarer, in the limit, infinitely rare. Neither of those means that the number of primes is finite.
They are "rare" in some senses, but not others. There are enough of them, for example, that the infinite sum of their reciprocals diverges.
You cannot conclude that p1*...pn+1 is prime. For example 2*3*5*7*11*13+1 = 30031, which is divisible by 59, hence NOT prime. However, what you CAN conclude is that p1*...pn+1 is either prime, or has a prime factor larger than any of the given pk's.
Because Jesus awoke after 3 days in a cave, and some heathens were cooking up some bacon just outside the chilly, dank tomb. The lord thy god Jesus Christ Almighty then smellethed the seared meat and thoughteth, "Mmmmm... Bacon." So now, every Christian may partaketh of the flesh of the filthy pig.
But thou must NOT consumeth uncooked tapeworms, for to do so makes thee unclean before the lord so that ye must follow the sin by a manual scouring of thy duodenum with an S.O.S. pad --- so as to maketh thyself pure again and worthy of the kingdom of heaven. Amen.
I'm only guessing here, so take it for what it's worth. Gizmo manufacturers come out with new models all the time. If the old model was reliable for a long time, they would put themselves out of business. I'm not saying all products are designed with a built-in expiration date (although that may be true sometimes). However, all products do in fact fail at some point. Almost surely, quality control has estimated a mean failure time, and adjust the time period covered by warranty accordingly.
Turning off a television is a far cry from shooting someone with a firearm. It is fucking hilarious how serious some of these posts are. "If you turn off my T.V. I'm gonna Kiiiillll You!" or "I'm gonna whup yore asss!"
Childish? Most of human behavior is childish --- from the kindergarten playground to the highest levels of academia, from the bedroom to the courtroom. Like it or not, it is a FUCK OR BE FUCKED world. Mountains of historical evidence support that statement.
It is a given that most people are going to be screwed over by their neighbors, their government, their employers, and probably even some of their friends, lovers, and family. Moreover, 9 times out of 10, there ain't a goddamned mother fucking thing they can do about it. You see, that's the way the system is set up.
So, if my choices are reduced to (1) eat shit, or (2) eat shit AND throw some back, I'll take the latter.
There are so many posts saying the same thing --- "What gives you the right?" I'll just pick yours;)
First of all, you are correct. No one has "the right" to turn off someone else's T.V. set. Just like no one has "the right" to bring their dog to the park to take a shit and leave the mess, along with 50 other pet owners. But it happens all the same.
No one gives inconsiderate cell-phone users "the right" to yap on it in the library where I'm reading a book. But it happens all the same.
No one gives people "the right" to break bottles and leave shards of glass strewn all over the beach. But it happens all the same.
Need I go on? This talk of "rights" is pointless. It's all about whose ox is being gored. I also am frustrated with the growing level of rudeness, noise, and inconsideration I encounter from hour to hour. If I can use some type of hidden device to, how shall I say it, "get even", then so be it. Sure I don't have "the right" --- but so what? If everyone else can have their "entertainment", then so can I.
Incidentally, as regards your comment about what the vast majority are fine with --- the vast majority are probably fine with smoking marijuana and driving 90 m.p.h. on the highway. But just let the cops catch you doing it.
The result is that the greatest asshole reaps the greatest bennefit. The people who do not value peace and quiet are never for want. Those who do not like it, have to distance themselves from the greatest assholes, leaving public spaces full of the most horrible people immaginable.
amen, I say AMEN, brother. The only problem is, we are rapidly running out of peaceful places to go. I only venture forth from my cave when absolutely necessary: to go to work, pick the kids up from school, buy groceries. I don't go out for fun anymore, because I always have to share my fun with 50-10,000 other individuals who smoke, throw trash everywhere, scream and yell, pick fights, leave their cell-phones on, bring their dog that defecates all over the place, you name it, whether I'm at at the beach or the public library. The crowds have ruined everything.
The idea would be that you could only black-ball one person in some (long) time interval, and that it would take several people black-balling the same person within the same time span to boot them off. If the troll didn't get enough black balls within the allotted time, the counter would be reset to zero. Maybe the black-ball tokens could be randomly distributed as well, so people couldn't form gangs, e-mail each other, and say "lets bounce this guy."
ANY system will get abused. Civilization has been around for what --- 10,000 years? Democracy, Monarchy, Communism, Anarchy --- you name it --- have been abused. Taking that as a given, a more realistic goal would be to design systems that try to minimize abuse.
With heavy moderation comes groupthink and censor of ideas that the group might not like.
Maybe not. If moderation was distributed randomly, rather than as an reward for "good behavior", with no person allowed to moderate for longer than a certain time interval, things might not be so bad. Pests could be banned by black-balling --- say everyone has 1 black-ball to spend per every 2 months (non-cumulative). Then, if some troll gets 10 blackballs within that time frame, they're outta there!
Maybe equal pay for all won't work. But, look at the current system. I've heard many capitalists say the same kind of thing --- there has to be a bottom and a top. Fine. But, you must then accept the social consequences. As the top shrinks to a point, and the bottom swells into the millions, the net result will be disintegration of the social fabric. Watch it happen before your very eyes.
Furthermore, do you think that all of the CEO's who get paid 6 and 7 digit salaries actually EARN that money? Was it all just hard work finally paying off? Or do these business magnates resort to lying, stealing, violence (by proxy) and bribing politicians to make laws favorable to their bank accounts? And if that's so --- and it is --- then as the saying goes: what's good for the goose is good for the gander. To make up for the tiny paycheck and reduction in personal freedom, the little guy is justified in stealing from the rich.
Our system is falling apart. Consumer debt is spiraling out of control. Our national debt is spiraling out of control. Do you think all of this debt is ever going to be paid? What happens when people/countries don't pay their debts?
So, you equate "equal pay for all" with "communism".
From Merriam-Webster, Communism -- 1 a: a theory advocating elimination of private property, b: a system in which goods are owned in common and are available to all as needed.
The flaw in your reasoning is that it assumes that none of the drugs currently under prohibition are worse than alcohol. This is simply not true.
How do you measure "worse"? Somewhere around 100,000 people die each year due to alcohol related illness. (Search the web for stats.) Factor in the hazard people pose to others while driving drunk, and the abused spouses, children, and friends of alcoholics, and we see that alcoholism collects an enormous toll on society --- a cost that dwarfs any ill effects caused by the relatively small number of people who do hard drugs.
If you've ever met a real cocaine or heroin addict (there is no such thing as a casual cocaine/heroin "user", only "addicts"),
Really? You mustn't get around much. I've known several. I used to do coke casually. I've never bought any, and I haven't used it in years.
you'd see that it is not the same thing as alcohol at all. It takes over your life. It creates a dysfunctional circumstance in which the person quickly loses the ability to carry on a normal life, hold a steady job, and maintain a loving relationship with family.
Alcohol does the same damage. So does addiction to gambling.
People can smoke cigarettes and still lead normal lives.
Yeah. Until they get lung cancer. Then it's time to sue the pusher man! Furthermore, the smokers get to take their friends and family down with them --- second hand smoke. You don't usually see a heroin junkie stabbing their wife and kids with a needle full of smack.
One last note. Show me where it says in the constitution of this fine country that the government has the right, indeed the duty to protect citizens from themselves? I seem to recall something about "life, liberty, and the pursuit of happiness." Surely, this must include the right to pursue one's own destruction.
I suppose you're right. To roll a six-sided die three times and have it come up a 4 each time is a --- what did you call it? oh yeah --- totally dumb selection of realistic data points. What was I thinking? That could never happen in the real world.
See the thing about averages is that there's a bottom, a middle, and a top.
Okay. What is the average of 4 and 4 and 4? Let's see... (whir)(click)(buzzz)(ching!)... The correct answer is 4, which is neither above nor below any of the numbers in the list. Thanks for playing. Try again.
In my opinion, the problem isn't cellphones so much as people being intolerant of other people.
In my opinion, the problem isn't people being intolerant of other people so much as people thinking they are the only individuals on the whole planet, and thus feel no qualms about being rude, inconsiderate, self-absorbed bastards.
On that front, I would suggest you look at Principia Mathematica by Russell and Whitehead, which is pretty much the book on purest mathematical foundations.
If you can afford to purchase a complete copy. I have to *56.
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Oops! Typo before the Q.E.D. should read: p' > p.
No, it is not equally dodgy. The fundamental theorem of arithmetic states that a composite number may be written (uniquely, apart from rearranging the factors) as a product of primes. Hence, the given number has a prime factor not in the list, contradicting the assumption.
I prefer a slightly different way of looking at the proof. Claim: The list of primes is infinite. This is equivalent to claiming that for any prime p, there exists another prime p' > p. Proof: Let q be the successor of the product of all primes up through and including p. Then, if q is prime, clearly q > p. If q is not prime, then by the fundamental theorem of arithmetic, it has a prime factor p'. But, p' cannot be any of the prime factors through p. Therefore, p' > q. Q.E.D.
In this way, we don't have a proof by contradiction.
That is true. Nevertheless, you made the additional claim that a certain number was prime, and used that FALSE claim to arrive at your contradiction. Your proof was incomplete, so I corrected it.
First, I assume you mean twin primes. Proving infinite primes is trivial and from ancient Greece. It is a proved fact that there are arbitrarily large gaps in the prime sequence (i.e. infinitely large gaps). And that primes get rarer and rarer, in the limit, infinitely rare. Neither of those means that the number of primes is finite.
They are "rare" in some senses, but not others. There are enough of them, for example, that the infinite sum of their reciprocals diverges.
That is a good one --- 211 = 13 * 16 indeed. :(
You cannot conclude that p1*...pn+1 is prime. For example 2*3*5*7*11*13+1 = 30031, which is divisible by 59, hence NOT prime. However, what you CAN conclude is that p1*...pn+1 is either prime, or has a prime factor larger than any of the given pk's.
Because Jesus awoke after 3 days in a cave, and some heathens were cooking up some bacon just outside the chilly, dank tomb. The lord thy god Jesus Christ Almighty then smellethed the seared meat and thoughteth, "Mmmmm... Bacon." So now, every Christian may partaketh of the flesh of the filthy pig.
But thou must NOT consumeth uncooked tapeworms, for to do so makes thee unclean before the lord so that ye must follow the sin by a manual scouring of thy duodenum with an S.O.S. pad --- so as to maketh thyself pure again and worthy of the kingdom of heaven. Amen.
People don't have to wait for an account anymore!
I'm only guessing here, so take it for what it's worth. Gizmo manufacturers come out with new models all the time. If the old model was reliable for a long time, they would put themselves out of business. I'm not saying all products are designed with a built-in expiration date (although that may be true sometimes). However, all products do in fact fail at some point. Almost surely, quality control has estimated a mean failure time, and adjust the time period covered by warranty accordingly.
is \int_{\-infty}^{\infty}e^{-x^2}dx = \sqrt{\pi} with the associated quotation from Lord Kelvin:
A mathematician is one to whom that is as obvious as that twice two makes four is to you. Liouville was a mathematician.
Turning off a television is a far cry from shooting someone with a firearm. It is fucking hilarious how serious some of these posts are. "If you turn off my T.V. I'm gonna Kiiiillll You!" or "I'm gonna whup yore asss!"
Childish? Most of human behavior is childish --- from the kindergarten playground to the highest levels of academia, from the bedroom to the courtroom. Like it or not, it is a FUCK OR BE FUCKED world. Mountains of historical evidence support that statement.
It is a given that most people are going to be screwed over by their neighbors, their government, their employers, and probably even some of their friends, lovers, and family. Moreover, 9 times out of 10, there ain't a goddamned mother fucking thing they can do about it. You see, that's the way the system is set up.
So, if my choices are reduced to (1) eat shit, or (2) eat shit AND throw some back, I'll take the latter.
There are so many posts saying the same thing --- "What gives you the right?" I'll just pick yours ;)
First of all, you are correct. No one has "the right" to turn off someone else's T.V. set. Just like no one has "the right" to bring their dog to the park to take a shit and leave the mess, along with 50 other pet owners. But it happens all the same.
No one gives inconsiderate cell-phone users "the right" to yap on it in the library where I'm reading a book. But it happens all the same.
No one gives people "the right" to break bottles and leave shards of glass strewn all over the beach. But it happens all the same.
Need I go on? This talk of "rights" is pointless. It's all about whose ox is being gored. I also am frustrated with the growing level of rudeness, noise, and inconsideration I encounter from hour to hour. If I can use some type of hidden device to, how shall I say it, "get even", then so be it. Sure I don't have "the right" --- but so what? If everyone else can have their "entertainment", then so can I.
Incidentally, as regards your comment about what the vast majority are fine with --- the vast majority are probably fine with smoking marijuana and driving 90 m.p.h. on the highway. But just let the cops catch you doing it.
The result is that the greatest asshole reaps the greatest bennefit. The people who do not value peace and quiet are never for want. Those who do not like it, have to distance themselves from the greatest assholes, leaving public spaces full of the most horrible people immaginable.
amen, I say AMEN, brother. The only problem is, we are rapidly running out of peaceful places to go. I only venture forth from my cave when absolutely necessary: to go to work, pick the kids up from school, buy groceries. I don't go out for fun anymore, because I always have to share my fun with 50-10,000 other individuals who smoke, throw trash everywhere, scream and yell, pick fights, leave their cell-phones on, bring their dog that defecates all over the place, you name it, whether I'm at at the beach or the public library. The crowds have ruined everything.
The idea would be that you could only black-ball one person in some (long) time interval, and that it would take several people black-balling the same person within the same time span to boot them off. If the troll didn't get enough black balls within the allotted time, the counter would be reset to zero. Maybe the black-ball tokens could be randomly distributed as well, so people couldn't form gangs, e-mail each other, and say "lets bounce this guy."
ANY system will get abused. Civilization has been around for what --- 10,000 years? Democracy, Monarchy, Communism, Anarchy --- you name it --- have been abused. Taking that as a given, a more realistic goal would be to design systems that try to minimize abuse.
With heavy moderation comes groupthink and censor of ideas that the group might not like.
Maybe not. If moderation was distributed randomly, rather than as an reward for "good behavior", with no person allowed to moderate for longer than a certain time interval, things might not be so bad. Pests could be banned by black-balling --- say everyone has 1 black-ball to spend per every 2 months (non-cumulative). Then, if some troll gets 10 blackballs within that time frame, they're outta there!
Just a thought. I'm not going to implement it.
Maybe equal pay for all won't work. But, look at the current system. I've heard many capitalists say the same kind of thing --- there has to be a bottom and a top. Fine. But, you must then accept the social consequences. As the top shrinks to a point, and the bottom swells into the millions, the net result will be disintegration of the social fabric. Watch it happen before your very eyes.
Furthermore, do you think that all of the CEO's who get paid 6 and 7 digit salaries actually EARN that money? Was it all just hard work finally paying off? Or do these business magnates resort to lying, stealing, violence (by proxy) and bribing politicians to make laws favorable to their bank accounts? And if that's so --- and it is --- then as the saying goes: what's good for the goose is good for the gander. To make up for the tiny paycheck and reduction in personal freedom, the little guy is justified in stealing from the rich.
Our system is falling apart. Consumer debt is spiraling out of control. Our national debt is spiraling out of control. Do you think all of this debt is ever going to be paid? What happens when people/countries don't pay their debts?
So, you equate "equal pay for all" with "communism".
: a theory advocating elimination of private property, b : a system in which goods are owned in common and are available to all as needed.
From Merriam-Webster,
Communism -- 1 a
You are batting two for two.
I was correcting his math, not his economic philosophy.
The flaw in your reasoning is that it assumes that none of the drugs currently under prohibition are worse than alcohol. This is simply not true.
How do you measure "worse"? Somewhere around 100,000 people die each year due to alcohol related illness. (Search the web for stats.) Factor in the hazard people pose to others while driving drunk, and the abused spouses, children, and friends of alcoholics, and we see that alcoholism collects an enormous toll on society --- a cost that dwarfs any ill effects caused by the relatively small number of people who do hard drugs.
If you've ever met a real cocaine or heroin addict (there is no such thing as a casual cocaine/heroin "user", only "addicts"),
Really? You mustn't get around much. I've known several. I used to do coke casually. I've never bought any, and I haven't used it in years.
you'd see that it is not the same thing as alcohol at all. It takes over your life. It creates a dysfunctional circumstance in which the person quickly loses the ability to carry on a normal life, hold a steady job, and maintain a loving relationship with family.
Alcohol does the same damage. So does addiction to gambling.
People can smoke cigarettes and still lead normal lives.
Yeah. Until they get lung cancer. Then it's time to sue the pusher man! Furthermore, the smokers get to take their friends and family down with them --- second hand smoke. You don't usually see a heroin junkie stabbing their wife and kids with a needle full of smack.
One last note. Show me where it says in the constitution of this fine country that the government has the right, indeed the duty to protect citizens from themselves? I seem to recall something about "life, liberty, and the pursuit of happiness." Surely, this must include the right to pursue one's own destruction.
I suppose you're right. To roll a six-sided die three times and have it come up a 4 each time is a --- what did you call it? oh yeah --- totally dumb selection of realistic data points. What was I thinking? That could never happen in the real world.
See the thing about averages is that there's a bottom, a middle, and a top.
... (whir)(click)(buzzz)(ching!) ... The correct answer is 4, which is neither above nor below any of the numbers in the list. Thanks for playing. Try again.
Okay. What is the average of 4 and 4 and 4? Let's see
In my opinion, the problem isn't cellphones so much as people being intolerant of other people.
In my opinion, the problem isn't people being intolerant of other people so much as people thinking they are the only individuals on the whole planet, and thus feel no qualms about being rude, inconsiderate, self-absorbed bastards.
As I thought ... they are full of piss.
On that front, I would suggest you look at Principia Mathematica by Russell and Whitehead, which is pretty much the book on purest mathematical foundations.
If you can afford to purchase a complete copy. I have to *56.
Personally, I prefer ZFC.