Hard Light Productions Forums
Off-Topic Discussion => General Discussion => Topic started by: Sandwich on July 09, 2002, 04:47:31 pm
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Ladies and Germs, I present: The Million Dollar Math Problem (http://www.extremetech.com/article2/0,3973,341723,00.asp)! Let the quasi-algebraical quantum hyperstringing begin! :lol: :wakka: :wakka: :lol:
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And we never saw CP again.....
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Oh this! The Riemann Hypothesis is probably the most famous and most important unsolved math problem today; I have heard of it many, many times, and have done some very basic investigations into it. Basically, the object is to prove why exactly all of the zeta function's non-trivial zeros (i.e. non-integral - the negative even numbers are also zeros) seem to lie on the critical line y=½+ix. Seems that the non-differentiable points (the points in question that equal zero) reach some sort of minimum for Re(x)=½, since values close to that, both higher and lower, do not reach zero. Extremely fascinating stuff; I might go into this field professionally later on. I would consider it a great achievement of humanity if anyone is able to prove this. :nod:
BTW the Riemann zeta function z(a) is defined as the following:
¥
å 1/xª
x=1
It has values for all numbers in the complex plane except for a vertical pole at a=1. This function has many, many cool properties and next to the gamma and hypergeometric pFq functions, is probably the most important special function around.
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Originally posted by Blue Lion
And we never saw CP again.....
already disproven, such a terrible hypothesis :p
(j/k btw)
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Originally posted by CP5670
Basically, the object is to prove why exactly all of the zeta function's non-trivial zeros......{incomprehensible mathematical stuff here}
Oh, thaaaat! That's easy: God made it that way. :D
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Sure, but why? :p
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Originally posted by sandwich
Oh, thaaaat! That's easy: God made it that way. :D
No I didn't :D
j/k here, people. no need to go ballistic.
Carry on with the incomprehensible mathstuff :)
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Originally posted by CP5670
Sure, but why? :p
He wanted to give you something to do. He likes maths people too you know. :)
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Originally posted by CP5670
Sure, but why? :p
Oh, come on - if He made everything relatively easy to solve/comprehend, what would all of the scientists do, twiddle their thumbs? (hmmm, interesting mental picture...)
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Yes, that is what I mean; let's find out why this hypothesis is true! :p :D
Here is something that the NYTimes article linked from there says: :D (G.H. Hardy was one of the experts in this field and one of my favorite mathematicians)
Hardy, wrote his biographer, Constance Reid, was convinced "that God — with whom he waged a very personal war — would not let Hardy die with such glory."
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Originally posted by CP5670
Yes, that is what I mean; let's find out why this hypothesis is true! :p :D
Here is something that the NYTimes article linked from there says: :D (G.H. Hardy was one of the experts in this field and one of my favorite mathematicians)
hehehe :)
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sandwich asked that I put a few screens up, so here's what the function looks like:
(http://www.3dap.com/hlp/hosted/procyon/misc/zeta1.gif)
x is going from 0 to 5 here. Looks a bit like the 1/x function here. Notice the pole at x=1.
(http://www.3dap.com/hlp/hosted/procyon/misc/zeta2.gif)
The negative portion of the above graph. x varies from -5 to 0, and the y value has been restricted between -0.01 and 0.03 to show the zeros at each of the even negatives.
(http://www.3dap.com/hlp/hosted/procyon/misc/zeta3.gif)
This is a plot of the function along the line x=1/2 and a variable imaginary portion. (x goes from 0 to 30i and y goes from -1.5 to 1.5) The important zeros can be seen here; note that they appear to be irrational.
(http://www.3dap.com/hlp/hosted/procyon/misc/zeta4.gif)
This is a 3D plot of z=|z(x+iy)|, with x going from 1 to 4 and y from 0 to 40. The depressions in the hills correspond to the x=1/2 line; the value of the function goes down to zero in each of those ditch-like areas. The x=1 pole is the black thing in the corner.
It is fairly easy to compute values for all positive even integers and negative odd and even integers - some examples are the following:
z(-2)=0
z(-1)=-1/12
z(0)=-1/2
z(2)=p^2/6
z(4)=p^4/90
z(6)=p^6/945
z(½+i 14.134725200497455...)=0 (this is one of those nontrivial zeros)
Unfortunately, analyzing anything other than these integers (excluding positive odds, which are also hard) is much more difficult.
And yes, I spend all day and night with this stuff. :D
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*shakes head*
Well, I guess I'm a visually-oriented person, considering that I "understood" those graphs far better and faster than I ever would have studying all that Greek stuff. :D
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I know! I know the answer! The answer is forty-two!
*thinks*
Damn. Desn't work. Oh well, time to continue searching...
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Originally posted by GalacticEmperor
I know! I know the answer! The answer is forty-two!
You never know, it could be
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You know, I watched a program on these people who solved these unsolvable math problems. Like that dude who solved Fermat's Last Theorem. It took him 7-8 years, tens of thousands of sheets of paper, the works of a Japanese mathematician who blew his brains out after giving up on his own crazy theories, and locking himself up in his room for the whole time. And in the end, he figured out an equation. He said it felt like eternal bliss, but I doubt his wife would say the same. :D
But the funniest by far is the story of that one guy who wrote 600 pages of equations to PROVE that 2+2=4. Damn. Some people just have a lot of time on their hands. I suggest we all just forget about math, and leave it to the calculators. :D
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I got a riddle that all of you (with a bit of experience) can solve. Its not CP's level of mathematical expertise :rolleyes: .
How can you prove that 1+1 equals 2?
(The awser is not in 1 line...;7)
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Originally posted by CP5670
x is going from 0 to 5 here. Looks a bit like the 1/x function here. Notice the pole at x=1.
[color=sky blue]Its been a while since I've done any math not related to chemistry or population studies so I'm a bit rusty. Am I correct in assuming it does the paradoxical thing of approaching in reducing increments as it gets closer to x=o so it never actually reaches the axis? [/color]
Originally posted by CP5670
This is a plot of the function along the line x=1/2 and a variable imaginary portion. (x goes from 0 to 30i and y goes from -1.5 to 1.5) The important zeros can be seen here; note that they appear to be irrational.
[color=sky blue]I've never seen anything like that. The oscillations are completely asymetrical then it nosedives. I think they may have found the God they were waging a personal war with...j/k[/color]
(http://www.uniquehardware.co.uk/server-smilies/contrib/ruinkai/cistinebiggrinA.gif)
Originally posted by sandwich
*shakes head*
Well, I guess I'm a visually-oriented person, considering that I "understood" those graphs far better and faster than I ever would have studying all that Greek stuff. :D
[color=sky blue]You're not alone. My gut reaction to that was to want to attempt graphing it for a better understanding but then I saw with great relief that graphs had already been posted. It wouldn't have been quite as pretty as those.[/color] :doh: :D
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well, thank god,now I finally have something to read when I have trouble sleeping.
well, time to stop spamming.
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Originally posted by Dark_4ce
You know, I watched a program on these people who solved these unsolvable math problems. Like that dude who solved Fermat's Last Theorem. It took him 7-8 years, tens of thousands of sheets of paper, the works of a Japanese mathematician who blew his brains out after giving up on his own crazy theories, and locking himself up in his room for the whole time. And in the end, he figured out an equation. He said it felt like eternal bliss, but I doubt his wife would say the same. :D
The thing they don't know is, the solution to Fermat's Last Theorem is either much simpler than that, or Fermat didn't really know the answer in the first place. The guy that solved it probably tried to work all his way around it using the constructs he knew to be proven (some even devised much after Fermat's death), while Fermat certainly realized a very simple answer for it, or was just putting senseless thoughts on paper. Mathematicians have a strong tendency to ignore common sense, you know. ;)
And I have a much better (and useful, in my opinion) problem for you to solve: Is P = NP? :)
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Originally posted by TheCelestialOne
I got a riddle that all of you (with a bit of experience) can solve. Its not CP's level of mathematical expertise :rolleyes: .
How can you prove that 1+1 equals 2?
(The awser is not in 1 line...;7)
N+N=2
N is equal to 1.
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Well, I guess I'm a visually-oriented person, considering that I "understood" those graphs far better and faster than I ever would have studying all that Greek stuff.
I am an analytic-oriented guy (just like Lagrange :D) so the equations mean much more to me than the graph. Although the greek letters do tend to get on my nerves. :p
I know! I know the answer! The answer is forty-two!
*thinks*
Damn. Desn't work. Oh well, time to continue searching...
Actually, in this case, we know what the answer probably is (all zeros are on x=1/2); it's just that nobody knows why it is so. :p
You know, I watched a program on these people who solved these unsolvable math problems. Like that dude who solved Fermat's Last Theorem. It took him 7-8 years, tens of thousands of sheets of paper, the works of a Japanese mathematician who blew his brains out after giving up on his own crazy theories, and locking himself up in his room for the whole time. And in the end, he figured out an equation. He said it felt like eternal bliss, but I doubt his wife would say the same.
Yeah, this guy Andrew Wiles has become quite famous among the mathematical community. Fermat's Last Theorem is pretty easy to understand - the equation x^n+y^n=z^n has no integer solutions for x, y and z when n is greater than 2. Proving it is another matter though. Wiles actually ended up uniting classical number theory and diophantine equations along with topology and elliptic curve theory - two of what were thought to be completely unrelated areas of mathematics - by partially proving the Taniyama-Shimura conjecture about elliptic curve modularity (Taniyama was the guy who shot himself after getting frustrated with a math problem :p) and showing that this would also prove the Last Theorem.
I got a riddle that all of you (with a bit of experience) can solve. Its not CP's level of mathematical expertise .
How can you prove that 1+1 equals 2?
(The awser is not in 1 line...)
As you might know, Whitehead and Russell attempted to tackle this problem with a 500-page proof derived from symbolic logic axioms in their monumental work Principia Mathematica. But there was an interesting ending to this; Gödel's famous theorem, published shortly after, showed that the consistency of mathematics is actually decidedly indeterminate, and thus it is impossible to create a consistent proof showing this. A sad result, but nevertheless a very important one.
[color=sky blue]Its been a while since I've done any math not related to chemistry or population studies so I'm a bit rusty. Am I correct in assuming it does the paradoxical thing of approaching in reducing increments as it gets closer to x=o so it never actually reaches the axis?[/color]
Exactly, the limits on either side of the x=1 asymptote are ¥ and -¥. Same kind of thing as with the y=1/x graph at x=0. ;)
[color=sky blue]I've never seen anything like that. The oscillations are completely asymetrical then it nosedives. I think they may have found the God they were waging a personal war with...j/k[/color]
Yeah, the zeta function does indeed have some very unusual properties. It is a periodic function but seems to actually have an infinite number of distinct periods - they change based on whether the nearby numbers are even or odd. Is the nosedive thing you are talking about the part at the left side of the image?
The thing they don't know is, the solution to Fermat's Last Theorem is either much simpler than that, or Fermat didn't really know the answer in the first place.
The fact that the methods used to prove the Theorem were almost all invented in the last 50 years has indeed raised speculation about whether or not Fermat actually had a proof (he was in the habit of scribbling results in his notebooks without publishing them, which is where this result was first found); most people think that he probably did not actually have a proof of it and thought it to be correct by trial-and-error, but nobody really knows, since he was a genius of the highest order at any rate.
Is P = NP?
It can be, if either P or N equals zero. :D
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Originally posted by CP5670
It can be, if either P or N equals zero. :D
Hmm, I think you know what I'm talking about, and it's not that... :p
To clarify: P is the set of problems with solutions that require an amount of time that is proportional in a polynomial relation to the size of the input, while NP is the set of problems with solutions that require an amount of time that is not proportional to a polynomial relation to the size of the input.
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Like that was readily obvious. :p I guess it would have no logically consistent solutions then, since a set operator x cannot equal ¬(x) by definition.
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Originally posted by CP5670
Like that was readily obvious. :p I guess it would have no logically consistent solutions then, since a set operator x cannot equal ¬(x) by definition.
Eh, you managed to miss the point again... :p
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Eh, you managed to miss the point again... :p
Was there one? :p Seems like a somewhat strange question anyway; asking for what set an operator and its counterstatement are equal... :p :D
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Originally posted by CP5670
Was there one? :p Seems like a somewhat strange question anyway; asking for what set an operator and its counterstatement are equal... :p :D
Nope, the point is to prove if there is an actual mathematical difference between the problems inside each of the sets or not. Our current knowledge of mathematics only allows us to do experimental tests on it, that are, per definition, inconclusive. There's also the fact that the only thing creating a difference may be our lack of knowledge of the ways to solve the (currently) NP problems in polynomial time.
Example: the knapsack problem can only be solved in a time that's proportional in an exponential way to the size of the entry - meaning that the processing time grows exponentially to the input size. But it may perfectly be because we do not know yet of the best way to solve it. Since most cryptography algorythms rely on the large times required to solve NP (normally NP-Complete) problems with large inputs, if the answer to the "Is P = NP" question is true, it may destroy every single cryptography system currently in use.
In short, the question is: are non-polynomial problems really non-polynomial or we just don't know enough of math as of yet? It's one of the most famous "unsolveable" problems of modern mathematics and computer science.
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Never mind what I said earlier; I read up a bit more on this problem in various other places and I now see what you are talking about. (actually I initially thought you were talking about PNP and NPN transistors :D) The two probably are equal at some point - new and sometimes surprising transformation identities are being discovered all the time - and besides, any mathematical theory would be incomplete without a synthesis uniting everything.
What is this "polynomial time" btw? Solving time that varies according to a polynomial expression of the input? If it is some sort of technical discrete mathematics or complexity theory term I probably will not know anything about it; all I know about is analysis and infinite process theory. :p
a much better (and useful, in my opinion) problem
Well this problem, while interesting, is not nearly as important or useful as the RH for us analytic number theory guys. :D
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As you might know, Whitehead and Russell attempted to tackle this problem with a 500-page proof derived from symbolic logic axioms in their monumental work Principia Mathematica. But there was an interesting ending to this; Gödel's famous theorem, published shortly after, showed that the consistency of mathematics is actually decidedly indeterminate, and thus it is impossible to create a consistent proof showing this. A sad result, but nevertheless a very important one.
Mmmm... I thought I had the solution... :
The proof starts from the Peano Postulates, which define the natural
numbers N. N is the smallest set satisfying these postulates:
P1. 1 is in N.
P2. If x is in N, then its "successor" x' is in N.
P3. There is no x such that x' = 1.
P4. If x isn't 1, then there is a y in N such that y' = x.
P5. If S is a subset of N, 1 is in S, and the implication
(x in S => x' in S) holds, then S = N.
Then you have to define addition recursively:
Def: Let a and b be in N. If b = 1, then define a + b = a'
(using P1 and P2). If b isn't 1, then let c' = b, with c in N
(using P4), and define a + b = (a + c)'.
Then you have to define 2:
Def: 2 = 1'
2 is in N by P1, P2, and the definition of 2.
Theorem: 1 + 1 = 2
Proof: Use the first part of the definition of + with a = b = 1.
Then 1 + 1 = 1' = 2 Q.E.D.
Note: There is an alternate formulation of the Peano Postulates which
replaces 1 with 0 in P1, P3, P4, and P5. Then you have to change the
definition of addition to this:
Def: Let a and b be in N. If b = 0, then define a + b = a.
If b isn't 0, then let c' = b, with c in N, and define
a + b = (a + c)'.
You also have to define 1 = 0', and 2 = 1'. Then the proof of the
Theorem above is a little different:
Proof: Use the second part of the definition of + first:
1 + 1 = (1 + 0)'
Now use the first part of the definition of + on the sum in
parentheses: 1 + 1 = (1)' = 1' = 2 Q.E.D.
:p
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Thunder must be so proud of CP5670 and Sandwich for promoting smart conversations on the forums :)
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Ok kids.... time for magazine time.... *looks around*
*feels really dumb about getting B in Jr. High Top Algebra Class*
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Def: Let a and b be in N. If b = 1, then define a + b = a'
(using P1 and P2). If b isn't 1, then let c' = b, with c in N
(using P4), and define a + b = (a + c)'.
Problem is, you are indirectly assuming that addition works here. What exactly defines x' in relation to x? :p
If you want to know more on why this cannot be done, you might want to check out Kurt Gödel's On Formally Undecidable Propositions of Principia Mathematica and Related Systems.
Thunder must be so proud of CP5670 and Sandwich for promoting smart conversations on the forums
hehe :D
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Ok... You got a point there... Can I ask you some more Q's from (wich I think :rolleyes: ) I have the solution?
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Originally posted by TheVirtu
Thunder must be so proud of CP5670 and Sandwich for promoting smart conversations on the forums :)
*waits to see about that...* :D
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Ok... You got a point there... Can I ask you some more Q's from (wich I think :rolleyes: ) I have the solution?
Sure, but I don't know if I will know the answers either. :p :D
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try to solve the following equation :
y'=a*y-b*y*y*y, y(0)=y_0
(I have the awnser... Unless you disprove it again... :rolleyes: )
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oh dear, eh ahem, cough cough, i must have lost my way, is this the Hard Light forum?
sorry for my ignorance, i am but a poor, ignorant renderer, lowly before you great mathematicians... :D
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Originally posted by CP5670
What is this "polynomial time" btw? Solving time that varies according to a polynomial expression of the input? If it is some sort of technical discrete mathematics or complexity theory term I probably will not know anything about it; all I know about is analysis and infinite process theory. :p
Simple explanation:
P, or "Polynomial Time" problems:
Processing time = (([input size]+1) * 4) seconds
NP, or "Non-Polynomial Time" problems (exptime and others):
Processing time = ([input size] ^ 3) seconds
Originally posted by CP5670
Well this problem, while interesting, is not nearly as important or useful as the RH for us analytic number theory guys. :D
Well, it's one of the most important ones for Computer Science (together with the state machine "stopping" problem - another "unsolveable" ones), and it affects any and all computer security systems of today. What are the practical uses of the problem that started this thread?
Ah, and "1+1=2" can be proved through formal semanthics. :D
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Everything Equals N its a fact!
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Simple explanation:
P, or "Polynomial Time" problems:
Processing time = (([input size]+1) * 4) seconds
NP, or "Non-Polynomial Time" problems (exptime and others):
Processing time = ([input size] ^ 3) seconds
Aren't they both polynomial operators in the algebraic sense though? ( 4(x+1) and x³ )
Well, it's one of the most important ones for Computer Science (together with the state machine "stopping" problem - another "unsolveable" ones), and it affects any and all computer security systems of today. What are the practical uses of the problem that started this thread?
Actually, the practical uses all occur in quantum theory, but seriously who cares about that? :D The important thing will be that the solution of the RH will reveal why the prime numbers are distributed the way they are, and thus solve many other fundamental problems in number theory.
Ah, and "1+1=2" can be proved through formal semanthics. :D
Eh...you are going to prove it through semantics (study of meanings in a linguistic sense) instead of symbolic logic? :wtf: :p Read that book I mentioned earlier; it has been proven that this task cannot be done and that mathematics is not an independently self-consistent system. (Gödel's incompleteness theorem)
y'=a*y-b*y*y*y, y(0)=y_0
Is this a differential equation? This one is first order and has no independents in the equation, so it should be fairly easy. Here's the solution:
dy/dx = ay - by³
dy/( ay - by³ ) = dx
ò dy/( ay - by³ ) = ò dx
(the integral on the left can be computed by splitting into 1/y and 1/(a-by²) and using integration by parts)
log(y)/a - log(a-by²)/2a = x + c
lim [ log(y)/a - log(a-y²)/2a ] = y0 + c
y->0
The limit will only be finite when a=0, and thus a must be 0 for a solution to exist.
c= -y0
Thus, x = y0 . (since the limit on the left goes to 0 when a=0 regardless of x)
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Originally posted by CP5670
Aren't they both polynomial operators in the algebraic sense though? ( 4(x+1) and x³ )
Err, actually, I made a little mistake there. It's not "Processing time = ([input size] ^ 3)", it's "Processing time = (3 ^[input size])". My bad. :p
Originally posted by CP5670
Actually, the practical uses all occur in quantum theory, but seriously who cares about that? :D The important thing will be that the solution of the RH will reveal why the prime numbers are distributed the way they are, and thus solve many other fundamental problems in number theory.
Bah, my problem is much cooler, and actually has practical applications in everyday's life. It 0wnZ your problem. :D
Originally posted by CP5670
Eh...you are going to prove it through semantics (study of meanings in a linguistic sense) instead of symbolic logic? :wtf: :p Read that book I mentioned earlier; it has been proven that this task cannot be done and that mathematics is not an independently self-consistent system. (Gödel's incompleteness theorem)
Argh, I won't go into specifics of formal semantics, but yes - you can prove it through symbolic logic applied to semantic constructs. Mathematical constructs are nothing but a very simple and objective language, and you can analyse it's semanthics as well (or even better, due to it's inherent simplicity) as any other language. And in the end, 1+1 will equal 2, whatever symbols you define for each of the elements.
The proof that you mentioned (that I didn't really look into) must have been tried using only mathematical constructs, and that indeed will not be possible. You need a higher level logic to analyse semanthics.
(note: I'm not even checking the equations on this thread, so don't point me to any of those things :D :p )
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An NP problem means that the validity of a possible solution can be performed in polynomial time. The actual, solution cannot be found in polynomial time.... it's improtant in making suppossably 'unbreakable' codes which can actually be verified, like in encryption.
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If you cant solve a problem, nuke it!
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Originally posted by Zeronet
If you cant solve a problem, nuke it!
So.... if you can't figure out how to get into your house when you've locked yerself out, blow the whole house up?
I thik I'll try that.... :devilidea
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Gosh - this has stayed on-topic! *stunned look on face*
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Originally posted by sandwich
Gosh - this has stayed on-topic! *stunned look on face*
Now, tell us the truth - what you really wanted was to have us all buggin' CP again, right? :D
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Originally posted by aldo_14
So.... if you can't figure out how to get into your house when you've locked yerself out, blow the whole house up?
I thik I'll try that.... :devilidea
Exactly!
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Err, actually, I made a little mistake there. It's not "Processing time = ([input size] ^ 3)", it's "Processing time = (3 ^[input size])". My bad.
I thought it might be something like that. ;)
Bah, my problem is much cooler, and actually has practical applications in everyday's life. It 0wnZ your problem. :D
Who gives a crap about practical applications? :p Hardy, the guy I mentioned earlier, once said that if the math is useful, then it is not worth learning - only useless math is good math - and I have some sympathy for this. :D Some of these mathematicians are on the brink of insanity due to a lack of knowledge on how the prime number distribution works. :D
Argh, I won't go into specifics of formal semantics, but yes - you can prove it through symbolic logic applied to semantic constructs. Mathematical constructs are nothing but a very simple and objective language, and you can analyse it's semanthics as well (or even better, due to it's inherent simplicity) as any other language. And in the end, 1+1 will equal 2,
whatever symbols you define for each of the elements.
The proof that you mentioned (that I didn't really look into) must have been tried using only mathematical constructs, and that indeed will not be possible. You need a higher level logic to analyse semanthics.
:wtf: Are you saying that human language is "higher level" than mathematical logic for analyzing such problems? :D Also, unlike some other topics out there (such as that of the cardinality of the real numbers), that proof I mentioned earlier is indeed globally accepted. Somewhere in the proof you probably assumed that addition works without knowing it; that has been the case with all such proofs out there. ;)
Incidentally, have you heard of Hilbert's famous 23 problems? The 8th one (the RH one) is really cool, but the 19th and 23rd ones are even better IMO. :D
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There are two current claims to the proof of the Riemann hypothesis. They're
here (http://www.coolissues.com/mathematics/riemann.htm)
and
here (http://www.coolissues.com/mathematics/zeta.htm)
And another big conjecture solved recently other than Fermat's last theorem is Catalan's conjecture that if you create a large table of numbers which are the powers of two (ie 4,8,16,32,64...etc), three (9,27,81, etc), four, five, etc, none of the numbers have a difference of exactly 1 apart from 8 and 9 (2^3 and 3^2).
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Those articles are both by the same guy; I looked through them and saw some rather strange assumptions and a few parts that were not really done right. (since when does Li(x) equal the integral of 1/x? :p) Thing is, with all of these types of problems, there have been (and still are) thousands of cranks out there that have claimed to prove it. :p :D
I have heard of the Catalan's cojecture as well; another interesting problem that was solved very recently, only about two months ago. Although this one is related more to classical number theory rather than analytic number theory, which is what I am into. :D
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OH, FOR THE LOVE OF GOD!!! :eek: :eek2:
I go away for 2 weeks, come back and find that MENSA have taken over the HLPBB. What's up with that? :wtf: :p
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See, now did anyone listen when I said MENSA were trying to take over the world? No. See what happens when you don't listen?
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I always thought it was the Freemasons...
That or the UN. Keep your story straight.:D
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Originally posted by Stryke 9
I always thought it was the Freemasons...
That or the UN. Keep your story straight.:D
But that's the problem! MENSA or the UN or the Freemasons have been twisting world events so expertly that we just can't tell what's going on anymore! :nervous: Who is the real enemy? To be honest, I just don't know. :shaking:
Beafraid. Trustnoone.
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*grabs flame-thrower*
Viva la revelution!
*goes to burn down MENSA*
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Originally posted by Stryke 9
I always thought it was the Freemasons...
Now, in all seriousness, that is a society that's truly frightening... :nervous: