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        riddles >> easy >> INFINITE PRIMES?
        
(Message started by: The_Fool on Jan 12th, 2004, 10:08am)
Title: INFINITE PRIMES?
Post by The_Fool on Jan 12th, 2004, 10:08am

Are there an infinite amount of primes?  As the numbers get higher, so does the space between the primes.  Therefore, as you approach infinity, you just can't get another prime.  Am I right or wrong, and how do you approach this?
Title: Re: INFINITE PRIMES?
Post by towr on Jan 12th, 2004, 10:30am
You're wrong.

It's also allready on the site I think.. But I'll repeat the argument (hidden)
::[hide]Start by supposing there are finitely many primes, then there is a highest numbered prime pmax.
Now multiply all primes up to and including that prime and add one.
You now have a number N that isn't a multiple of any of the known primes, and thus has to have k distinct new prime factors (where 1 <= k < N/pmax).[/hide]::
Title: Re: INFINITE PRIMES?
Post by TenaliRaman on Jan 12th, 2004, 10:51am

Quote:
As the numbers get higher, so does the space between the primes.


Interestingly there is a conjecture which says that "there are infinitely many twin primes. This conjecture directly contradicts your statement.
Title: Re: INFINITE PRIMES?
Post by towr on Jan 12th, 2004, 11:05am
You have to consider, a conjecture is just that, conjecture, not theorem. (It would be nice if we could find a proof for it)

The prime density does seem to decrease (it approximately follow 1/ln(N), which for N -> [infty] is 0) So the average distance also decreases.
But all this just means the probability some number is prime is 0 for large numbers, not that they don't exist. 0 probability events do occur, every day even..
Title: Re: INFINITE PRIMES?
Post by TenaliRaman on Jan 12th, 2004, 11:13am
towr,
I understand what you mean.
Its just the statistical data accumulated so far gives a strong support to this conjecture.(i don't mean to say that it is enough .. i hope u get what i mean).

If i get bored of life some day i will just sit and try to prove this conjecture and not to miss the riemann hypothesis as well.  ;D
Title: Re: INFINITE PRIMES?
Post by James Lu on Jan 13th, 2004, 8:30am
towr,

What are examples of 0 probability events that occur?
Title: Re: INFINITE PRIMES?
Post by Sameer on Jan 13th, 2004, 8:40am
The event that you will ever post in this forum  ;D
Title: Re: INFINITE PRIMES?
Post by towr on Jan 13th, 2004, 10:54am

on 01/13/04 at 08:30:46, James Lu wrote:
towr,
What are examples of 0 probability events that occur?
-Any event that only occurs once, for example. Or more generally any event that occurs a finite number of times, where there are an infinite amount to choose from.
-Selecting a real number with uniform distribution from an open interval. Or more generally with any continuous distribution.
-When moving from some point A to a point B, the probability you're at any specific point in between A and B.

There are also cases where two probabilities are 0, but when you divide them you get a very reasonable normal probability.. For instance with a conditional probability, where you look at a finite subset of an infinite set.
f.i. if you move through the open interval <0, 10>,
the probability of being on an integer, P(integer), is zero
the probability of being on an integer that is divisible by three, P(integer and div3), is also 0,
but the probability of being on a number divisible by three given that you are on an integer,  P (div3|integer) = P(integer and div3)/P(integer), is 1/3.
Title: Re: INFINITE PRIMES?
Post by Sir Col on Jan 13th, 2004, 4:14pm
In my understanding, the problem with probabilities is that they are only defined over exclusively countable (discrete) or exclusively measurable (continuous) domains for which the process of selection is clearly defined. It is meaningless to talk about the probability of selecting a discrete value from a continuous set. For example, P(selecting 2 from [0,4]) is undefined. Firstly, the process of selection is not possible, and secondly, 2 and [0,4] are incompatible elements. Whereas P(selecting [1.5,2.5] from [0,4])=1/4, and has meaning.

However, the greatest philosophical challenges lie with examples such as the probability of rolling 7 on an ordinary 6-sided die, or spinning a head or a tail with an ordinary coin. Is it reasonable to talk something in terms of probabilities if there is no measure of likelihood?
Title: Re: INFINITE PRIMES?
Post by SWF on Jan 13th, 2004, 6:08pm
Using The_Fool's reasoning, there are a finite number of integers that are pefect squares:

As the numbers get higher, so does the space between perfect squares.  Therefore, as you approach infinity, you just can't get another perfect square.
Title: Re: INFINITE PRIMES?
Post by Icarus on Jan 13th, 2004, 6:11pm
The concept of probability lends itself easily to more general mixtures of discrete and continuous situations, but to do so adds another level of abstraction to the mathematics, which most students of probability are not ready for. So in courses discrete and continuous probabilities are kept chastely apart, lest virgin minds be polluted by their carnal mixing.

However, the probability of a discrete event in a continuum is well-defined in any case. For continuums unsoiled by the intrusions of discrete influences, the probability of any discrete event will always be zero. This means that any calculation involving them will either result in 0, or 0[cdot][infty], which is undefined. To define the values needed, you have to look at intervals. That is why in probability problems involving continuums you find everything expressed in terms of intervals or other sets with interior.
Title: Re: INFINITE PRIMES?
Post by John_Gaughan on Jan 13th, 2004, 7:17pm

on 01/13/04 at 18:08:55, SWF wrote:
Using The_Fool's reasoning, there are a finite number of integers that are pefect squares:

As the numbers get higher, so does the space between perfect squares.  Therefore, as you approach infinity, you just can't get another perfect square.


This reminds me of something I read about irrational numbers. I think it was a web page or book about random numbers, and it talked about [pi]. There are an infinite number of digits in [pi], and in those digits you will find every subsequence of digits. For example, somewhere in the digits of [pi] is a sequence of a million ones in a row. And this is to be expected. This is more random number theory but still is related to numbers and infinity so leave me alone :-)
Title: Re: INFINITE PRIMES?
Post by THUDandBLUNDER on Jan 13th, 2004, 8:49pm

Quote:
For example, somewhere in the digits of [pi] is a sequence of a million ones in a row.

Only one sequence?   :o
Title: Re: INFINITE PRIMES?
Post by Eigenray on Jan 13th, 2004, 10:56pm
Here's something to think about when picking real numbers "at random."

Define an equivalence relation on [0,1) by saying x is equivalent to  y iff x-y is rational.  Then the equivalence class of each x represents a countable set of real numbers, and there are therefore an uncountable number of equivalence classes.

Now, use the Axiom of Choice, and form a set S by taking one element from each equivalence class.

Pick a real number from [0,1) at random.  What is the probability it lies in S?

Remember, we like probability measures to be
(1) Countably additive.  If S1, S2, ..., is a countable collection of disjoint sets, then the probability that x lies in their union should be the sum of the probabilities of x lying in each set.
(2) Translation (rotation?) invariant.  The probability that x lies in S should be the same as the probability that it lies in S+t = {x+t mod 1 | x in S}, for any t.
Title: Re: INFINITE PRIMES?
Post by towr on Jan 14th, 2004, 12:59am

on 01/13/04 at 16:14:17, Sir Col wrote:
It is meaningless to talk about the probability of selecting a discrete value from a continuous set.
Not really, it is simply an integral of the probability density function over a domain containing one element, and thus zero.
It is the same as the probability of selecting a value within a certain interval, only the interval in the discrete case is just [v], rather than [a,b]. Integrating over [a,b] is equivalent to using <a,b>, and for [v] = <v> = {} it gives 0.
Using the distribution function (which is the primitive of the probability density function), for an interval [a,b] you get D(b)-D(a), and for [v] you get D(v)-D(v) = 0, perfectly well defined and logical.


Quote:
However, the greatest philosophical challenges lie with examples such as the probability of rolling 7 on an ordinary 6-sided die, or spinning a head or a tail with an ordinary coin. Is it reasonable to talk something in terms of probabilities if there is no measure of likelihood?
I don't see why that is a philosophical challenge. Since there isn't a 7 on an ordinary 6-sided die you logically can't role it. If somehow you do role a 7, then you need only remember that falsum implies everything, so you may make a wish.
Mathematics is very forgiving with the absurd in a way.. It's really not that judgemental..
(Of course there's a zero probability that due to quantummechanics one of the eyes from the other five sides displaces to the six eyed side, in which case you may role a seven, so it's not really impossible ;))
Title: Re: INFINITE PRIMES?
Post by John_Gaughan on Jan 14th, 2004, 6:33am

on 01/13/04 at 20:49:03, THUDandBLUNDER wrote:
Only one sequence?   :o


Actually I said "there is a sequence," as in [exists]. Not "one sequence." I suspect there are an infinite number of each sequence but I'm not a mathematician so I'll stop short of trying to prove it.
Title: Re: INFINITE PRIMES?
Post by Benoit_Mandelbrot on Jan 14th, 2004, 8:35am
Well, since there is no limit to numbers, you will eventually find another prime, because if you give me the highest prime you can come up with, I can find another number above that, until I find another prime, so there should be an infinite number of primes.
Title: Re: INFINITE PRIMES?
Post by John_Gaughan on Jan 14th, 2004, 8:52am

on 01/14/04 at 08:35:35, Benoit_Mandelbrot wrote:
Well, since there is no limit to numbers, you will eventually find another prime, because if you give me the highest prime you can come up with, I can find another number above that, until I find another prime, so there should be an infinite number of primes.

I remember seeing a proof somewhere that was either inductive or used a similar process that proved there are infinite primes. Of course with larger numbers primes are spread out more, but you can always find a larger one.

Someone mentioned here or in another thread that prime number distribution decreases logarithmically, so they get more sparse, but in a predictable (but not absolute) way. Since the logarithmic curves increase to infinity (however slowly), is there ever a prime number Pn with an infinite number of composites between it and the next prime Pn+1, or would this effectively say that Pn is the last prime (which is false)?

Maybe one of the math PhDs here could englighten us. This was the original question but thread got hijacked early on ;-)
Title: Re: INFINITE PRIMES?
Post by Dudidu on Jan 14th, 2004, 9:10am

on 01/14/04 at 08:52:05, John_Gaughan wrote:
Maybe one of the math PhDs here could englighten us. This was the original question but thread got hijacked early on
John_Gaughan hi,
Maybe this proof (http://www.utm.edu/research/primes/notes/proofs/infinite/euclids.html) (from 300 BC) will help you.
Title: Re: INFINITE PRIMES?
Post by towr on Jan 14th, 2004, 9:16am

on 01/14/04 at 08:52:05, John_Gaughan wrote:
I remember seeing a proof somewhere that was either inductive or used a similar process that proved there are infinite primes.
You can read my version of it in the second post from the top.


Quote:
Someone mentioned here or in another thread that prime number distribution decreases logarithmically, so they get more sparse, but in a predictable (but not absolute) way.
in the fourth post :p
a better approximation is the li function. There are about li(n) primes among the first n postive whole numbers


Quote:
Since the logarithmic curves increase to infinity (however slowly), is there ever a prime number Pn with an infinite number of composites between it and the next prime Pn+1, or would this effectively say that Pn is the last prime (which is false)?
The latter, while not entirely meaningless (you should hear Icarus on this) transinfinite numbers are a hard concept to wrap you mind around, more importantly they aren't integers, so they can't be prime anyway.

Most of these things are also (better) explained in the following mathworld pages
http://mathworld.wolfram.com/PrimeNumberTheorem.html
http://mathworld.wolfram.com/PrimeNumber.html
Title: Re: INFINITE PRIMES?
Post by Icarus on Jan 14th, 2004, 7:12pm
To be a bit more explicit, transfinite numbers do not form a domain, so the concept of "prime" is not defined for them. I.e., multiplication on transfinite numbers does not allow the concept of "prime".

However, I don't think John meant to delve into the transfinite realm with his question, but rather was caught by a common misunderstanding of natural numbers.

Any natural number has only a finite set of other natural numbers less than it, so it is impossible for an infinite set of natural numbers to lie between any two. In particular, any two consecutive primes can only differ by a finite amount.

And to reiterate towr's argument in the 1st reply of this thread(which is a reiteration of Euclid's argument), there are infinitely many primes (each prime is finite, but there is always a next higher prime). Choose any finite set of primes. Take the product of all the primes in that set, then add 1. Finally factor the result back into primes. None of the primes in this second list is from the original set, since dividing the result by any of the primes in the original set always leaves a remainder of 1.

Hence, for any finite set of primes, there are other primes not in the set. So the set of all primes must be infinite.
Title: Re: INFINITE PRIMES?
Post by John_Gaughan on Jan 14th, 2004, 7:35pm

on 01/14/04 at 19:12:09, Icarus wrote:
However, I don't think John meant to delve into the transfinite realm with his question, but rather was caught by a common misunderstanding of natural numbers.

Well of course I did not mean that since I don't know what transinfinite numbers are. But I figured there was something that might explain it in the higher mathematics.
Title: Re: INFINITE PRIMES?
Post by Icarus on Jan 15th, 2004, 3:36pm

on 01/14/04 at 19:35:32, John_Gaughan wrote:
Well of course I did not mean that since I don't know what transinfinite numbers are. But I figured there was something that might explain it in the higher mathematics.


Transfinite (or [i] infinite[\i], but not "transinfinite") numbers are numbers greater than or equal to infinity. These numbers are not a part of the Natural numbers or the Reals, or the Complex numbers even. There are many ways in which such numbers can be defined. I described three of the most common in this post (http://www.ocf.berkeley.edu/~wwu/cgi-bin/yabb/YaBB.cgi?board=riddles_medium;action=display;num=1027804564;start=125#135) in the 0.999... thread. Another common one is a variation of the continuum infinities mentioned that attaches a single infinity to the complex plane which is the limit of anything whose magnitude increases without bound. This turns the complex plane into the "Riemann Sphere", a very useful tool in Complex Analysis. A better description can be found in the Complex Analysis forum.

However, I have never seen a set of infinite numbers with the appropriate properties for primes to exist.
Title: Re: INFINITE PRIMES?
Post by John_Gaughan on Jan 15th, 2004, 9:38pm
Ah, I misread the "transfinite" part. Either way it is a new concept to me. Thanks for the link. I don't understand half of that post but that's okay, I'm getting there. I really do love math. Between this forum and wikipedia I am learning more than I ever thought existed in math. Some day I hope to learn more about math. Hmm... I guess this is some day :-)
Title: Re: INFINITE PRIMES?
Post by Barukh on Jan 16th, 2004, 12:25am

on 01/14/04 at 08:52:05, John_Gaughan wrote:
...is there ever a prime number Pn with an infinite number of composites between it and the next prime Pn+1...

Interestingly enough, Euclid's argument about the infinitude of primes may be applied to show that for any finite n - no matter how big - there exist n consecutive composite numbers: just take (n+1)! + 2 to (n+1)! + (n+1).
Title: Re: INFINITE PRIMES?
Post by Icarus on Jan 18th, 2004, 11:35am

on 01/15/04 at 21:38:35, John_Gaughan wrote:
Ah, I misread the "transfinite" part.


No you didn't: towr used the wrong word in his post - I was just trying to set the record straight as to which is the correct usage. It doesn't matter that much though. There is nothing that I am aware of that is officially called transinfinite, so the meaning is clear to anyone familiar with these numbers anyway.

If you are interested in learning more about the behavior of the infinite, the puzzles "Impish Pixie", "For the Honor of Hufflepuff", and "Transinfinite Subway" in the Hard forum are mind-bending challenges: Particularly the latter, which relies on behavior so weird that even I (though having much experience with infinite numbers) have a hard time accepting what logic demands.

The thread "Denumerability Dilemma" in the "General Problem Solving" forum also discusses aspects of the theory of cardinal numbers. Some others do as well.
Title: Re: INFINITE PRIMES?
Post by THUDandBLUNDER on Jan 18th, 2004, 12:03pm

on 01/15/04 at 15:36:40, Icarus wrote:
Transfinite (or [i] infinite[\i], but not "transinfinite").


on 01/18/04 at 11:35:03, Icarus wrote:
If you are interested in learning more about the behavior of the infinite, the puzzles "Impish Pixie", "For the Honor of Hufflepuff", and "Transinfinite Subway" in the Hard forum are mind-bending challenges.
1) The latter thread is entitled 'Transfinite Subway'.  ::)

2) The initial post with the question has mysteriously disappeared, as have 60 odd other of my posts.
(Lost to posterity for ever?)  :'(

Perhaps this is what James Fingas has been up to recently (I've not seen him around for a while) - he has been writing himself a backdoor!
:D

Title: Re: INFINITE PRIMES?
Post by towr on Jan 18th, 2004, 12:24pm

on 01/18/04 at 11:35:03, Icarus wrote:
No you didn't: towr used the wrong word in his post
damn, I knew I should've looked it up, rather than just guess.. :-X
Title: Re: INFINITE PRIMES?
Post by Icarus on Jan 18th, 2004, 1:06pm

on 01/18/04 at 12:03:28, THUDandBLUNDER wrote:
1) The latter thread is entitled 'Transfinite Subway'.  ::)

D'oh!

:-[ :-[ :-[ :-[ :-[ :-[
Title: Re: INFINITE PRIMES?
Post by John_Gaughan on Jan 18th, 2004, 11:29pm

on 01/18/04 at 11:35:03, Icarus wrote:
If you are interested in learning more about the behavior of the infinite, the puzzles "Impish Pixie", "For the Honor of Hufflepuff", and "Transinfinite Subway" in the Hard forum are mind-bending challenges: Particularly the latter, which relies on behavior so weird that even I (though having much experience with infinite numbers) have a hard time accepting what logic demands.

The thread "Denumerability Dilemma" in the "General Problem Solving" forum also discusses aspects of the theory of cardinal numbers. Some others do as well.

Now my brain hurts, thanks a lot! I guess I don't know as much about infinity as I thought...
Title: Re: INFINITE PRIMES?
Post by Sir Col on Jan 19th, 2004, 3:51am
I suppose that with infinity there's an awful lot to understand. ;)
Title: Re: INFINITE PRIMES?
Post by Sameer on Jan 19th, 2004, 6:51am
If i know correctly no one understand infinity "perfectly" in this world...
Title: Re: INFINITE PRIMES?
Post by Benoit_Mandelbrot on Jan 19th, 2004, 8:40am
The thing is that we can never know exactly what happens at [infty] because it isn't an actual number.  The best we can do is predict behavior there.  Sometimes, we have such a good idea like f(x)=x that as x -> [infty] so does f(x) with an accuracy of 99.999999999...%.  With PrimePi(x)-ln(x), the behavior can be erratic, and we don't really know what happens toward [infty].  Does it settle down, or go crazy?  Does the number of primes suddenly increase dramatically with extreme numbers?  We don't have the power yet.
Title: Re: INFINITE PRIMES?
Post by Sameer on Jan 19th, 2004, 9:25am
Let us shift the origin to (infinity, infinity). In that case the original origin (0,0) is transformed into an infinity point. Thereby the behavior of any graph at (0,0) would essentially be same as that on infinity but hard to explain in language of mathematics!!!  ::)
Title: Re: INFINITE PRIMES?
Post by Icarus on Jan 19th, 2004, 5:24pm

on 01/19/04 at 08:40:11, Benoit_Mandelbrot wrote:
The thing is that we can never know exactly what happens at [infty] because it isn't an actual number.


[Sigh] Here we go again. Please read through Misconceptions (http://www.ocf.berkeley.edu/~wwu/cgi-bin/yabb/YaBB.cgi?board=riddles_medium;action=display;num=1027804564;start=225#249) (4), (5), (6) in the 0.999... thread.

Bottom line, infinity is an actual number (more accurately, there are infinitely many actual numbers that are all infinite). But it is not a Real number (as in the set of Real numbers), or a complex number.


Quote:
The best we can do is predict behavior there.


No, we can do better than that. Just like everywhere else, we can logically deduce the behaviors of infinities from the definitions. This same means that tells us what happens at 1 and 2 and 3, also tells us what happens at [infty] and [smiley=varaleph.gif]0 and [omega].

The trick is not to assume that the nice familiar behavior of the finite numbers automatically applies to the infinite ones. This is a mistake that even experienced math users often make.


Quote:
Sometimes, we have such a good idea like f(x)=x that as x -> [infty] so does f(x) with an accuracy of 99.999999999...%.


I suppose this is technically correct, assuming that the 9s go on forever, so that this is exactly 100%. But the idea being presented here, that we are not entirely sure, is not true. The concept of limit is well-defined, though that definition is often not presented during a first introduction to calculus. By means of this definition, we can say with complete assurance that, for example, limx[to][subinfty] 1/x = 0, or as in your example, limx[to][subinfty] x = [infty].

(I do not blame you for this miscomprehension. It is a unfortunate bi-product of how mathematics is taught at your level. To understand the concepts behind mathematics requires a sophistication that most people studying mathematics do not begin to pick up until their sophomore or junior year of college. In order to develop that sophistication, it is useful to first introduce a less rigorous version of mathematics that politely ignores the deeper subtleties.)


Quote:
With PrimePi(x)-ln(x), the behavior can be erratic, and we don't really know what happens toward [infty].  Does it settle down, or go crazy?  Does the number of primes suddenly increase dramatically with extreme numbers?  We don't have the power yet.


The problem here is not with infinity but with [pi](n) (the number of primes less than n). In the mathworld links towr gives above, you can find the result, by Chebyshev, that

7/8 < [pi](n)ln(n)/n < 9/8.


This is a fairly strong result, and says that if [pi](n)ln(n)/n does not converge, it doesn't miss doing so by much.

But it is the nature of primes that make this a hard task, not the nature of infinity.


on 01/19/04 at 09:25:16, Sameer wrote:
Let us shift the origin to (infinity, infinity). In that case the original origin (0,0) is transformed into an infinity point. Thereby the behavior of any graph at (0,0) would essentially be same as that on infinity but hard to explain in language of mathematics!!!  ::)


Alas, that too would require that [infty] be a real number. But the geometric behavior of points at infinity is the subject of projective geometry, a branch of mathematics owing much of its development to artists, who studied it during the Renaissance in order to learn how to draw perspective views properly. Alas as well, once they developed the techniques, those alone were passed on to their descendants, so once again dunderheads can call themselves artists. :'(

Projective geometry represents the first move ever made by mathematicians beyond the safe confines of 3-dimensional Euclidean geometry. (It was viewed as an extension of Euclidean geometry, which is why the concept of non-Euclidean geometries still had to wait for Lobatchevsky, Bolyai, and Gauss a century or two later.)
Title: Re: INFINITE PRIMES?
Post by Sameer on Jan 20th, 2004, 6:50am
Icarus, that makes me wonder about relative velocities too ... ok here how it goes.. lets shift origin by 1 unit. Of course we can do that since it is a real number. Lets repeat this many many times... (read infinity?) Of course then we are only moving 1 unit relatively but ultimately reaching a point of infinity?

That was one of the things that boggled my mind about speed of light. How to find the boundary of universe. Let's say we can see very very well so that we see the boundary at a distance equivalent to speed of light. However any point at that boundary will see another boundary at a distance of speed of light (twice the speed of light?  :o). Ugh my mind is now hurting.

From practical point of view I know that if a system converges or diverges (that is at least of the states and non - oscillatory) I can safely assume the behavior at infinity to be practically same as within some larger multiple. As for e.g. if I give a step input to a resistor capacitor system (damped) then i know i can consider time t = 5*Tau (time constant) to be infinity for this system.
Title: Re: INFINITE PRIMES?
Post by rmsgrey on Jan 20th, 2004, 9:25am
The trouble with velocity is that relative velocities aren't additive (under Special Relativity etc) - If I'm travelling at speed x relative to the ground, and you're travelling at speed y relative to the ground in the opposite direction, then I measure your speed, z, as slightly less than x+y relative to me. Until you get up pretty close to the speed of light, the difference between z and x+y is pretty close to zero, and actual measurements will usually miss it, but it is there (according to our current understanding of the universe, and assuming it's large enough to escape quantisation problems)

If you start with 0 and add 1 to it an infinite number of times, you'll end up with something infinite, but there's still a discontinuity there - while you add 1 at each step, there is a point where you have no previous step - when you've just turned infinite - which is enough to start me off on a headache.

The definition of limits is such that, for practical applications, once you get far enough through a process, you are effectively measuring the value at infinity (the error in measurement is much greater than the difference in value) - because the idea of a limit is that for any arbitrary degree of precision you care to specify, there's a point beyond which all future measurements will be the same to that degree of precision. Or, for things that diverge to infinity, for any chosen threshhold value, there will always be a point beyond which the value never drops below that threshhold.
Title: Re: INFINITE PRIMES?
Post by Icarus on Jan 20th, 2004, 4:29pm
I am also unclear on what you mean by the "boundary of the universe". While it is possible that our universe has a spacial boundary, there is no evidence that it does, and most cosmological theories assume it does not. How can the universe be of finite size without a boundary? A 2-dimensional analog should make it clear: The sphere is a finite 2-dimensional object without boundary.

A better example though would be a cylinder, because there is considerable evidence of a past temporal boundary. Our universe is definitely of finite age. And the same curling around that is allowed spacially is not allowed by relativity in the temporal direction.
Title: Re: INFINITE PRIMES?
Post by Sameer on Jan 21st, 2004, 9:39am
I think I am also unclear cos I had read that few years back. This was somewhat related to Big Bang which states our universe is expanding and so should be of finite size. Also it gives us a paradox (?) that a point beyond this hypothetical boundary is travelling at a speed greater than speed of light (as mentioned by that relativity concept) I don't want to think about it as it hurts my head. :(

Now rmsgrey's post has started hurting my head in +1 step too... thinking thinking...

[Boss] Sameer are you working?
[Sameer] uh... yea yea.. trying to solve a problem  ;D
Title: Re: INFINITE PRIMES?
Post by Icarus on Jan 22nd, 2004, 3:26pm
That would be the limit of what we can see, which is another matter entirely. If the universe expands faster than the speed of light, which inflationary theory claims, then two particles that were formerly within causal contact can be separated so that neither can see the other. The claim, as I understand it, is that as long as it is because spacetime is expanding between them, it is okay for two particles to separate faster than light. I don't see it mathematically myself. It seems to me that you still have some observers seeing energy levels become infinite. But I haven't explored it, so I cannot gainsay the claim.

In any case, "border of the universe" is not quite right, because the two particles are still in the same universe. Places separated from us in this fashion are forever out of our influence now, however.
Title: Re: INFINITE PRIMES?
Post by Sameer on Jan 22nd, 2004, 4:14pm
If speed of the light is ultimate limit then what we can see would be the "instantaneous boundary" ... What matters would be - speed of light maybe ultimate limit for us, but is it ultimate limit for particles we don't know yet? Of course I know if we start discussing universe here this board will go on and on.. so i will stop it here unless someone wants to take it further..

my real point of giving that analogy was if we slowly move in measurable moves, after a "large amount" are we not at inifinity "relative" to where we were initially? SO in a way I deduce that if "infinity" is a known entity "but is large or infinite relative to the quantity in consideration" we can predict its behavior. However unmeasurable infinites are hard to predict? Again a contradictory example would be converging or diverging infinite series...

ok so now my already hurting head is hurting more and its too cold outside....  :'(
Title: Re: INFINITE PRIMES?
Post by Icarus on Jan 22nd, 2004, 7:06pm

on 01/22/04 at 16:14:39, Sameer wrote:
If speed of the light is ultimate limit then what we can see would be the "instantaneous boundary".


But this does not represent the boundary of the universe. The concept of "universe" must be defined more broadly to be meaningful. It might be described as everything that at some time in its history had an effect that we can determine. If these luminally-separated particles started off not so separated, then they had an effect we can determine. If they didn't ever, then there would have been no reason to assume their existance. So the universe is not necessarily limited to what we can see, even it Einstein's speed limit is now sacrosant.


Quote:
my real point of giving that analogy was if we slowly move in measurable moves, after a "large amount" are we not at inifinity "relative" to where we were initially?


Not unless our "large amount" is itself infinite. Which, assuming that our speed is finite, would require an infinite amount of time, an so we would never arrive.


Quote:
SO in a way I deduce that if "infinity" is a known entity "but is large or infinite relative to the quantity in consideration" we can predict its behavior.


Large is not the same as infinite by any means. Infinite is beyond large. We can predict the behavior of the Large by analogy, since large is only in reference to something else. But infinite is a designation that is absolute, not relative. And we can never see it in real life, since to do so would require infinite resources, which we do not have. The only way to approach the infinite is with logic.

But by the same token, since infinities are not experienced in the real world, infinity is nothing more than a logical concept. Far from being not understandable, in essence we created them, so we know exactly where to look when problems crop up - in the logic of their definitions.
Title: Re: INFINITE PRIMES?
Post by John_Gaughan on Jan 22nd, 2004, 9:19pm

on 01/22/04 at 19:06:18, Icarus wrote:
But by the same token, since infinities are not experienced in the real world, infinity is nothing more than a logical concept. Far from being not understandable, in essence we created them, so we know exactly where to look when problems crop up - in the logic of their definitions.

What about an event horizon? Isn't that a way of experiencing the infinite? I think of it like a limit -- as something gets closer and closer to a black hole, it gets slower and slower, infinitely slow... or does it?

In response to Sameer -- while nothing can travel faster than light that we know of, in a way, things can. Gravitational effects are instantaneous, for example. And there is debate about tiny "gravitron" particles that "implement" gravity. Whether gravity functions as matter or energy, it is instantaneous, or infinitely fast -- no matter what, it is faster than light.

And what about quantum physics? If energy were to travel through a quantum wormhole (I'm not talking scifi stuff here, I mean the ones that we know exist but are at the subatomic level) and appear instantaneously somewhere else, it might travel faster than the speed of light by our stationary observations. This does not mean it travels faster than light, since it takes advantage of a hole or "short cut" in space-time, but by our observation, it departs point A and arrives at point B, for a net speed of "faster than light."
Title: Re: INFINITE PRIMES?
Post by Sir Col on Jan 23rd, 2004, 12:28am

on 01/22/04 at 21:19:56, John_Gaughan wrote:
In response to Sameer -- while nothing can travel faster than light that we know of, in a way, things can.

Actually, you'll be pleased to know that physicists have been able to make light pulses exceed the speed of light for quite a long time now. Recently an Italian team were able to send microwaves at about 5-7% faster than c.

The limiting "law" is related to the causality principle, which basically says that cause must precede effect, and if something travelled faster than light we would observe the effect before the cause.

I'm no authority in this field, but I also believe that the limiting equations prohibit objects travelling at the speed of light, as mass increases towards infinity as they approach this speed. However, there is nothing wrong, theoretically, with objects travelling faster than the speed of light.
Title: Re: INFINITE PRIMES?
Post by towr on Jan 23rd, 2004, 1:21am

on 01/22/04 at 21:19:56, John_Gaughan wrote:
Gravitational effects are instantaneous, for example. And there is debate about tiny "gravitron" particles that "implement" gravity. Whether gravity functions as matter or energy, it is instantaneous, or infinitely fast -- no matter what, it is faster than light.
Recent experiment have shown gravity travels with the speed of light, and not faster.. It is not instantanious..

Berkeley Lab Physicist Challenges Speed of Gravity Claim (http://www.scienceblog.com/community/article1755.html) (I recall reading more extensive, later, articles about it in the paper as well..)


on 01/23/04 at 00:28:48, Sir Col wrote:
Actually, you'll be pleased to know that physicists have been able to make light pulses exceed the speed of light for quite a long time now. Recently an Italian team were able to send microwaves at about 5-7% faster than c.
As I recall the pulse as a whole still travels slower, but it's just the front of the wave that travels faster.


Quote:
I'm no authority in this field, but I also believe that the limiting equations prohibit objects travelling at the speed of light, as mass increases towards infinity as they approach this speed. However, there is nothing wrong, theoretically, with objects travelling faster than the speed of light.
It's just a bit hard to imagine, since their mass will be imaginary, and consequently their impuls as well..
1-v2/c2 will be smaller than 0 when v is greater than c, so
m = m0 / [sqrt](1-v2/c2) will be an imaginary number
Title: Re: INFINITE PRIMES?
Post by John_Gaughan on Jan 23rd, 2004, 6:09am

on 01/23/04 at 01:21:14, towr wrote:
Recent experiment have shown gravity travels with the speed of light, and not faster.. It is not instantanious..

Berkeley Lab Physicist Challenges Speed of Gravity Claim (http://www.scienceblog.com/community/article1755.html) (I recall reading more extensive, later, articles about it in the paper as well..)

I don't have the link handy but I read an article rebutting these claims. I don't know either way, I've read articles on both sides of the fence that present substantial evidence. It's like the old argument of whether light is matter (photons) or energy (EM wave), or maybe both. My point was just to present the possibility -- that this is being debated among physicists today.
Title: Re: INFINITE PRIMES?
Post by rmsgrey on Jan 23rd, 2004, 7:10am
I suppose it may have been resolved by now, but for quite a while, it was seriously suggested that neutrinos may have imaginary rest mass.

The laws of physics as we know them prohibit anything from accelerating to light speed (from above or below) and also anything with zero rest mass is prohibited from existing except while travelling at light speed. In general, the closer something gets to light speed (relative to an observer), the more mass the observer measures it as having.
Title: Re: INFINITE PRIMES?
Post by Icarus on Jan 24th, 2004, 8:42am
First of all, in a relativistic universe nothing can ever be "infinitely fast", and the concept of "instaneous", as you use it, is not even definable. Two events that occured at the same time from the perspective of one observer will occur at different times from the perspective of other observers in motion with respect to the first. There is no frame of reference that is in any way superior to the others, so any concept of "simultaneousness" is dependent on the frame used.

So no, gravity cannot be instaneous without violating the constancy of the speed of light. It is also unlikely that gravitons travel faster than light, though the evidence here is much weaker. What one observer sees as FTL (faster than light) travel, to other observers is traveling backwards in time. If gravitons did this, we should see gravitational effects of a fast moving object precede the object in its path. i.e., if we were to deduce where the object was from it's gravitational affect on other objects, the place we would predict would be in front of where the object is actually located, as determined by other means. No such affect has ever been observed and verified.

Concerning the "light FTL" phenomena, this has been known for many years, assuming it is the one I am aware of, and towr appears to be refering to. It was a textbook item when I was in college. The math predicting it dates back to the early days of QM. It is the result of quantum tunneling (which despite the name has nothing to do with quantum wormholes). And towr is right, only the front of the photon's wave packet moves FTL. As the photon passes through the barrier, some of its energy is reflected, causing its wavelength to increase. This spreads out its wave packet, which means that the front of the packet extends forward FTL, the rear of the packet moves STL, so that the packet as a whole can travel exactly at the speed of light. You cannot detect just the front of the packet, so you cannot transmit any information FTL this way.

No one is arguing anymore whether the photon is a particle or a wave. It has been conclusively demonstrated to be both, as has all matter. And all matter is energy, so there is no debate over whether it is "matter or energy". These are subjects that were settled in the first half of the 20th century. The development of QM showed that the old "particle" vs "wave" division of the means for energy transference does not in fact exist. All waves are particles and all particles are waves. Instead we discovered a different division: Fermion vs Boson. Fermions make up the stuff that we normally think of as matter, Bosons the stuff we tend to think of as waves or, by great abuse, "energy".

There is no debate over which the photon is, or any other known particle. The behavior of fermions and bosons are widely divergent, so it doesn't take much to determine which a particle is. Photons are bosons.

Concerning the "imaginary mass" idea for neutrinos: In the 70s or early 80s, some experiments aimed at finding the rest mass of the neutrino produced results suggesting they traveled at a speed slightly greater than light. Everyone doubted that this was the case, but if the data were right, the implication would indeed be an imaginary mass, exactly as towr has stated. Further refinements have since shown that the original result were flawed, as everyone, including the researchers who produced those results, expected. The last I heard, evidence was mounting that neutrinos travel slower than light, with the possibility of them traveling >= c nearly, but not completely, ruled out.

Concerning the event horizon. John is correct that at the event horizon, some things become infinite. I had overlooked this in my statement. However, I do not believe that this can ever be measured, so using it a tool to experimentally explore the nature of infinities is not possible. I am fairly sure of this, but have not explored it enough to know definitely.
Title: Re: INFINITE PRIMES?
Post by Benoit_Mandelbrot on Jan 26th, 2004, 8:32am
The thing is the graviton is only theoretical.  We humans haven't actually proven it's existance yet.  It would be nice if we did prove it's existance, because we can then have fun with different kinds of gravitational anomalies, and find many easier ways to get into space and move about.  Once in orbit, we could capture gravitons from the sun, earth, or any other planet.  We could do it from the planet's surface.

If string theory is correct, and there are strings holding this universe together, we won't need to fly around space.  We can just modify a brane that joins the two parts of space together, and use it like a bridge.

The first thing I would do is to try to prove or disprove the existance of gravitons.  If there are gravitons holding everything together with gravitational energy, that would mean that the anti-particle anti-graviton would do the opposite.  One theory on why we don't see massive explosions from particles and there anti-particle is because they do everything opposite from their partner, even timewise.  They are traveling  through time differently than normal particles, with normal being the particles that we see every day.  The anti-photon would be something to see.

If everything started from one big moment and we set that as zero time, the anti-particles could be going in negative time.
Title: Re: INFINITE PRIMES?
Post by towr on Jan 26th, 2004, 8:49am

on 01/26/04 at 08:32:29, Benoit_Mandelbrot wrote:
If there are gravitons holding everything together with gravitational energy, that would mean that the anti-particle anti-graviton would do the opposite.
It doesn't.. As theory goes, the graviton is it's own anti-particle, two gravitons can annihilate each other, just like ..

Quote:
The anti-photon would be something to see.
.. just like photons do
At least this means you can see them ::)
Title: Re: INFINITE PRIMES?
Post by Benoit_Mandelbrot on Jan 26th, 2004, 10:01am

on 01/26/04 at 08:49:46, towr wrote:
It doesn't.. As theory goes, the graviton is it's own anti-particle, two gravitons can annihilate each other, ...

If this is true, then it could be very difficult to store them.  This could also be a reason why we can't really detect them, unless some recognizable form of energy is present.
Title: Re: INFINITE PRIMES?
Post by towr on Jan 26th, 2004, 10:31am
I don't see why that of all things would make it a problem in detection.. We can detect photons without a problem.
Storing would be a problem, but we don't generally try to store photons for use either. I would sooner look into a way to generate them.
Title: Re: INFINITE PRIMES?
Post by Icarus on Jan 26th, 2004, 3:49pm
The problem with detecting gravitons is that their predicted properties place them beyond the edge of our current capacities. The is little doubt as to their existance, but if they don't exist, then we need to figure out what else that spin 2 boson can be, or else abandon quantum field theory. Since QFT has proven highly successful so far, that does not seem likely.

No particle that travels at the speed of light has an anti-particle (or it is it's own anti-particle). Since the graviton is expected to travel at c, it falls under this rule. To understand why, you need to realize that at c, time does not exist. To a photon or graviton, time stands still. Its entire existance is but a single moment, and the entire universe is a single plane it exists in the middle of. Anti-particles can be thought of being the same particle, but traveling the opposite direction through time (an analogy that works mathematically, but should not be made too much of). Since time stands still for light-speed particles, there is no such distinction.
Title: Re: INFINITE PRIMES?
Post by John_Gaughan on Jan 27th, 2004, 5:32am

on 01/26/04 at 10:31:27, towr wrote:
I don't see why that of all things would make it a problem in detection.. We can detect photons without a problem.
Storing would be a problem, but we don't generally try to store photons for use either. I would sooner look into a way to generate them.

When I read your post I was thinking about a gravitron flashlight. The first thing that popped in my head was Mortal Kombat, where the guy grabs the other guy with his grappling hook or whatever and says "get over here!"

If we could generate gravitrons that would be incredibly cool. Why fuss over traveling to Mars when we could aim a gravitron cannon at it and pull it into our orbit, making things so much easier?
Title: Re: INFINITE PRIMES?
Post by THUDandBLUNDER on Jan 27th, 2004, 5:38am
And what does all this stuff have to do with Infinite Primes? ???
Title: Re: INFINITE PRIMES?
Post by John_Gaughan on Jan 27th, 2004, 5:51am

on 01/27/04 at 05:38:44, THUDandBLUNDER wrote:
And what does all this stuff have to do with Infinite Primes? ???

Well let's say you substitute prime numbers with gravitrons, composite numbers with photons, and 1 with an electron. Are there an infinite number of gravitrons?

You're right, nevermind ;-)
Title: Re: INFINITE PRIMES?
Post by Icarus on Jan 27th, 2004, 6:00pm
The infinite prime question was quite adequately answered in the first reply. That being so, there is no reason not to let the discussion go wherever it takes us!
Title: Re: INFINITE PRIMES?
Post by Sameer on Jan 28th, 2004, 7:44am

on 01/27/04 at 05:38:44, THUDandBLUNDER wrote:
And what does all this stuff have to do with Infinite Primes? ???

Heehee I think I would consider myself the culprit for this discussion!!!  ;D
Title: Re: INFINITE PRIMES?
Post by towr on Jan 28th, 2004, 12:05pm
here's an interesting article, "Faster-than-light shout seen echoing through space" (http://www.scienceblog.com/community/modules.php?name=News&file=article&sid=2208)
a small quote: "The European Space Agency's X-ray observatory, XMM-Newton, has imaged a spectacular set of rings which appear to expand, with a speed a thousand times faster than that of light"

(it's just a visual effect though ;)
Title: Re: INFINITE PRIMES?
Post by Benoit_Mandelbrot on Jan 29th, 2004, 8:23am

on 01/27/04 at 05:32:59, John_Gaughan wrote:
If we could generate gravitrons that would be incredibly cool. Why fuss over traveling to Mars when we could aim a gravitron cannon at it and pull it into our orbit, making things so much easier?

How would we put it back?  That would throw everything off!  Disasterous!  Mars is actually pretty boring!  We could use the graviton cannon on a satellite and pull the satellite to the planet!  We could do that since it has little mass compared to a planet!   We could create our own little black hole to study!
Title: Re: INFINITE PRIMES?
Post by John_Gaughan on Jan 29th, 2004, 8:30am

on 01/29/04 at 08:23:22, Benoit_Mandelbrot wrote:
We could create our own little black hole to study!

Maybe if we pulled enough mass toward Earth we could turn it into a black hole and finally see what's on the other side of an event horizon. We would probably die in the process, but it would be cool since it's in the name of science!
Title: Re: INFINITE PRIMES?
Post by rmsgrey on Jan 29th, 2004, 8:51am
At least for a first attempt at industrial scale gravity manipulation we'd probably want to use a small black hole to do it anyway, so creating another would be pretty redundant.


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