ElijahJones

Hopefully we can get past the semantics a little bit now that we know each other better. Yes, f:N->N where f(n)=p_n is a function but it doe snot produce prime numbers. In fact is'nt it true to say that it might be useful as a logical construct but in real life because we cannot derive and algortihm for its computation (like we can with say g(x)=x^2) it is of no practical use at all? Here is my idea on that x^n=z^n-y^n. We ar enot talking function we are talking map. On the left see a representative of the set {x^n|n>2, x an integer} on the right see a representative of the set {z^n-y^n|n>2, y, z integers} An assumption of a solution to Fermat says that for at least one pair of numbers in those sets the equation holds. If we can now show one solution implies infinitely many perhaps we can deduce reductio ad absurdum or some other contradiction. The point of the factorization I did on the left was to suggest that we could look at the monoids(?) formed by adding multiplication to the sets I listed above. It's a dead end but I don't think the reasoning was as ridiculuous as you made it out to be. By the way the issue of sequences and factorization is very interesting to me that's why I posted that factorization thread. You of all people should find something worth commenting on there.

You lost me on that one bro. I thought it was equivalent to notY => notX (the contrapositive). You must mean that symbol to be greater than or equal to. So x>=y has logical compliment x<y. I agree with that but I don't see how you can turn an order operation into a Boolean, the ordering implies more structure than a boolean algebra can account for does it not?

I had no idea it had gone so far, follow some of those links and you find like 30 different types of cardinals. I like the argument though that you cannot form the set of all sets because it would itself be a set you did not include. Its a nifty little logical circle from one perspective, but I never really saw how it was necessarily a problem because every set is considered a subset of itself. So the set of all sets by definition should already contain itself.

It could be a cover. The crew may have been replaced by alien doubles! No but if you were to try to place a metal battleship in the middle of a very strong time dependent field you are just asking for trouble. And I am not sure of the physics behind that, how does an EM field effect sonar or radar blips? I don't think the Germans had radar, the British developed it in 1944 if I am correct. It would be great if it were that simple to start teleporting things I doubt it though. And I am an EM fanatic, in fact I have quite an interst in ion propulsion within the earths atmosphere and using EM fields is one way to get alot of ions.

Not meaning to go spiritual but i am in that kind of mood. So when the Bible says that so and so lived for 969 years that is possible?

I'm not surprised, but nobody has yet mentioned that we may not be capable as a species of knowing the answer to certain questions. When we talk about causes that precede the big bang, how do we trace that deterministically through a singularity (if that was in fact what was there, there are some various scientific theories about that I guess). Personally I think that God may have used evolution, its an elegant method to be sure. I have heard Christians oppose this perspective because it seems unholy. But what do mere mortals know about holiness? Having this viewpoint leaves me in a very awkward position, a creationsist who believes in evolution and most of modern cosmology. In fact I think that Genesis is really a hodge podge of oral tradition. The law was given to Moses on the mountain, I don't see anywhere where Genesis was dictated to him by God or that anyone can interpret Genesis 1-11 accurately enough to rule out an old-earth. Can anyone? So anyways I think it just does not matter as much as we want it to. And I am certain that if I were God I would think they were confused who did not heed my words about when or how the earth was made. But I can't see how it becomes an issue of salvation. Unless of course they were using their "ideas" to tell people not to believe in me. I'm not God (thank God) but I doubt He would care as much about such things as He would about how we treat one another and (by the New Testament) whether we believe in Jesus. Atonement that comes through Christ seems a very archaic notion to us but it was hot stuff in 33AD, they all knew what atonement was because all of the religions had aspects of atoning sacrifice.

Thanks I'll chek it out. On that dna issue, Spencer Wells has found a mutation on the male gene that allowed him to track males. Its quite fascinating here is a link. http://www.itconversations.com/shows/detail255.html

First of all Luke noone can prove the universe is infinite (show me the professional reference, Journal Article?). Secondly Thales life cannot be inevitable unless it is one hundred percent certain, but the existence of natural variation shows that nothing is one hundred percent certain or have you missed the last one hundred years of science where they discovered the random basis for the universe. I agree that your analogy seems to make sense. Suppose I have a die and I roll it. I will wager that I will eventually roll 2 million fives in a row. Because the laws of probability say I must eventually roll every finite combination of numbers from the die, it is inevitable that someday I will roll 2 million fives in a row, it may happen in the first two million rows. But the probability of the event is only (1/6)^2,000,000. Its so small that on average you should expect to wait much longer than the universe has currently existed. You cannot tell me that that is inevitable. That life is I agree, but that it is inevitable you cannot prove that. It is probable and it has happened here for sure and likely elsewhere thats the best we can say and its likely the best we will ever be able to say for certain. Oh yeah I almost forgot. If the Big Bang "explains" where we came from where did the Big Bang come from? Saying that science explains things is rather dubious don't you think. Realizing that some mechanism lies behind the one you first observed is not an explanation of the phenomenon its simply a greater knowledge of what is. We hail certain discoveries because they are useful, but some we just idolize for no apparent reason. The Big Bang theory has many groupies but few masters.

Prenote: There is not way to take a function of the natural numbers and produce only primes. I am not playing semantics here. Hell I can prudce a function that will give lots of primes but not consecutively. In fact Dedekind showed that ax+b cannot be used to generate only primes for any a and b. you know who Dedekind was right? And before you get off and running on your debunking campaign you should know that I have a degree in mathematics. What? Who? By the way at least two of x and y being coprime is a way of getting primitive solutions. Saying that only n=3,4,5 are primitive is pointless, none of the solutions for those cases are general solutions to the problem, they are in effect a waste of time. Are you able to reproduce Wiles argument? I probably could but I can't find a full exposition of it anywhere. So as something to do that involves elementary number theory I started this thread. Furthermore equality of two expressions containing variables is a map. But I am sure that dogmatically I should only see an equation. But if in fact it has any solutions at all it can be thought of as a map between sets. Also wikipedia is a public resource edited by the public and especially on technical topics it is often wrong. Or do you think that they would breach national security on cryptography (as an example) to allow someone to post the latest breakthroughs? I can't help it if you cannot follow the line of argument, but I do have a degree from an accredited university (BS Math/Physics 2000 UWEC) and am about to finish my masters in Environmental Science and Policy with emphasis on resource management and mathematical modeling. My thesis is on mathematical modeling of ecological succession. After that I plan to apply to a Phd program in applied mathematics to study stochastic dynamical systems with applications to environmental modeling. Before I accept from you the old mathematical who? how? what? line of arrogance maybe you should list some credentials. I looked on your website and found none. Please tell me you are not an overread sophomore geek who thinks wikipedia is an encyclopedia of mathematics?

If I were to predict and I am usually wrong. I would guess that Sharon is going to feel compelled to negotiate quickly the fate of the West Bank settlements in exchange for compromise over Jerusalem. I think the realistic person on both sides realize that Jerusalem will have to be shared and that concessions in the West Bank on both sides might grease the wheels a bit. Also there is an issue of a secure highway between the West Bank and Gaza, Palestinian air and sea access. Those issues must be addresed so the Palestinitan people can become a part of the world market and begin developing infrastructure. In the last five years Israel has destroyed most of the infrastructure so the economy is sorely lacking. Once they start getting a little prosperity and a reprieve from violence on both sides the peace loving will grow strong. At some point their desire for a civil society will immediately criminalize any group paracticing terror or insiting it. That's the peace process my friends in a nutshell. I think that Abbas will follow through though he sometimes panders to Hamas. One good sign is that Hamas is going to take part in the upcoming elections but I expect trouble from them for many years to come. On the Israeli side Sharon is on a new course but I think Shimon Peres is the man to broker the final deal. Netanyahu is not a man to increase hostilities but he has a dogmatic view about the settlements, a view that Sharon has realized is untenable.

Well I know you guys closed out one of these but the concept is interesting. Ordinal numbers and cardinal numbers I know a bit but not everything and cardinal numbers are part of one of the most important math discoveries of the 20th century Paul Cohen's result on the continuum hypothesis. Any takers?

In circuits we worked with cos(x)+isin(x). The link for that sin(x+iy) is reputable I;m not sure why the expanded function the showed woul dbe called the complex sin function unless there is acoplete suite of six complex trig functions taken over the real valued trig and hyperbolic trig functions.

If you know what a manifold is you must already have some problem solving skills. You can't get to that level in math without knowing something about proofs. Do you know the logical fallacies? Also if you pick a subject in any of the sciences and study everything about it including all the current literature ofmr the journals you eventually see where the cutting edge is. At this point the subject shows you the outstanding problems (open questions). Number Theory is agood place to make problems for yourself. Pick a Diophantine Equation and find all of its solutions. Be careful it might take th rest of your life depending on what equation you choose. Here I will challenge you with one. Create your own problem by picking a and b and as any integers you like and then solving this Diophantine Equation. [MATH]ax+by^3=1[/MATH] Have fun and please know better than to mention to your frined that you slept with her wife. EJ

Spooky. Being quite simplistic I would have thought that it would arise if a human were picking form any set they percieved to be ordered. They feel closer to the smallest element so subconsciously they "balance" their random choices in kind. Is it a cognitive bias or is it supposed to be a real physical phenomenon?

There changes have been made. Well Fermat's Last Theorem stipulates that n>2. Yes you can parametrize the n=2 solution thats pretty nifty, its pythagorean triples. The point is that x cannot be prime under the construction below (that is really the question that needs to be proven or not I think and by me too, I am passing it off on anyone). If it is true how does that imply I can generate infinitely many primes from an elementary function? Thats already been shown impossible as far as I know. This thread is just for throwing around some ideas, its not intended to be the return of Frenchy's proof. So yeah that is right for course being able to show that neither x nor y is prime means they are composite and coprime (we say relatively prime over here in the States). What I think is interesting is that this maps {x^n}->Z[y,z]. {x^n|n is a non negative integer} is closed under multiplication and contains 1, it does not have inverses. I have not had time yet but I am hoping that this map will give some further insight into the problem. But you should be able to say something about that Matt if your research is in algebraic topology.

Dude you are'nt even doing calculus one things right in that. Look at what I did down there. The real deal here surrounds two questions. 1) Have you actually had a course in ordinary differentail equations? 2) Where did you come up with this equation. If you want my reference text it is Stewart, "Calculus 3rd Edition" and Boyce and Deprima "Elementary Differential Equations and Boundary Value Problems, 5th Edition". What you are doing with those differentials and the integrating factor suggests you do not know what you are doing to me anyways. Changing differentials mid stream requires calculations like this ds^2=dy^2+dx^2. thats how you get the differential for finding the length of a curve in the plane. Now if you want me to take you seriously give me some information about your training in math.

Well if you had the damn solution all along then why post here? If oyu had said it was a challenge people would have still responded. Anyways, I am not going to spend alot of time looking over what you have there I just did an analysis of the original equation which can be simplified very easily. So what exactly was your point? To try to show us all the new tool you have in your toolbox? Not too mention the fact that when you integrate 2 holding x constant you have to get the same function. What text did you get that equation from? Cuz I'm not here to waste time trying to make you feel like a mathematician. There are several steps in your derivation that are not obvious. Like how did your differential suddenly become de^(xy) and then in the next step you are back to something in dx. You cannot assume that the integrand was constant with change in e^xy. Show me some credentials or a text reference and I will take you seriously.

Hi Matt, I'm new and looking at dates is not a habit yet. Thanks EJ

e^(xy)(ycosx-sinx)+xe^(xy)cosxdy/dx=0 Well first of all I see no point in trying to develop exact methods for non-linear ODE's or PDE's when you can get as close as you want to any solution using numerical methods. But I did find exact equations in Boyce. Is it not clear that dy/dx=[-e^(xy)(ycosx-sinx)]/[xe^(xy)cosx] dy/dx=[sinx-ycosx]/[xcosx]=(tanx-y)/x I don't see where this amenable to any of the methods associated with exact equations. You might be able to use the implicit function theorem to get. dy/dx=(-F[x]/F[y])=(tanx-y)/x This yields F[x]=(y-tanx) F[y]=x If I integrate each of these with respect to their variable I get: integrating F[x] by x -> F=yx-int(tanx)dx 'note the integral of the tangent is not an elementary function it is an infinite series' integrating F[y] by y->F=yx. Not knowing where you got this equation I won't assume but if your limits of integration cause int(tanx)dx to vanish you could replace the function you quoted with F(x,y) = xy+C. That means that you would integrate on limits that are always [MATH]k\pi$[/MATH] apart.\ Ok, well wait a minute lets abck track and look at this. dy/dx=(tanx-y)/x y+xdy/dx=tanx I think you could get a power series solution here. Assume [MATH]y=\sum_{i=0}^{infinity}C_ix^i[/MATH] Then [MATH]xdy/dx=\sum_{i=0}^{infinity}iC_ix^i[/MATH] Giving [MATH]\sum_{i=0}^{infinity}C_ix^i + \sum_{i=0}^{infinity}iC_ix^i=tanx[/MATH] A little simplifying and [MATH]\sum_{i=0}^{infinity}C_i(1+i)x^i=tanx[/MATH] Now find a suitable power series representation for tanx (taylor series comes to mind) and equate coefficients.

You seem a little defensive Thales for a Greek who has been dead for over two millenium. My point is that getting bounds for the possible number of intelligent species is not as easy as simply multiplying a bunch of assumptions together. I to think it is likely that intelligent life exists elsewhere and I am convinced it is possible for the same reasons you mention. But I would caution against saying that life is a foregone conclusion (unless you are so tense about creation that you have to attack straw arguments to ease your conscience). In a universe that is filled with random variation nothing is a foregone conclusion, some things are likely but nothing is 100% guaranteed. The idea that the universe is progressing deterministically seems to be shattered by Quantum Mechanics. On the other hand if the universe is deterministic then it must have had a primary cause? What was this cause? Emergent properties? I study complex systems using mathematical models and an emmergent property is simply a possibility inherent in the structure of the system. There are an infinite number of system trajectories that evade various attractors in all but the simplest of systems. So if you are saying that human life was inevitable (which in some way suggests that we are very important in the underlying structure of the universe) I think your argument contradicts itself. If we are accidental then we are not a foregone conclusion and if we are not an accident, who or what set the process in motion? And please lets not say the Big Bang. I believe in the Big Bang. That does not explain where we or anything came from. So you like the idea that there are others out there, and probably like to think that we will someday meet them. If it makes your sense of the universe more complete then good for you. The thing I find interesting about the description of aliens thats common now is that it sort of got started with Roswell. Its shows strong cultural patterns, in fact I'm pretty sure there were movies before Roswell that depicted big eyes and pale skin. If aliens have been visiting regularly over the eons then each culture would likely have its own description and rationality for them. Our description is very thoughful if you think about it, because after having spent all of there lives for many generations on interstellar travel a big head with big eyes and very slender bodies would be one possible trajectory of adaptation. Its kind of spooky really. The little saucers, check into EM effects and you see that with enough power those might just be a form of landing craft. Combine that with their physical form and the mother ship must be stationed somewhere nearby,....perhaps behind the moon! That's why Neil Armstrong has acted so wierd. Did you know he is a senator now and could in position to vote to repeal NAFTA and lead the wordl down a slippery slope of environmental degradation leading to a planet only imhabitable by the aliens. Damn! Where is Charlie Sheen when you need him? No Luke, the one man 60K ago was genetically the father of us all, but he was not the only one living, check out Spencer Wells Phd. on the web at National Geographic. I wish you guys would lighten up a little! BTW Thales what data are you using to make your prediction?

Your question is a little peculiar because a strange attractor is an attractive fixed set that contains chaotic trajectories of a system of differential equations. These are continuous systems, however you can get discrete systems that are chaotic the quadratic map x->x^2+c is a great example. But to think of primes as being on a strange attractor you would need 1) A space containing the prime or whose set of limit points was the set of primes. 2) An interative function on that space. Now your last question about the largest distance between two primes is settled by the prime number theorem. Check it out! Eventually if you go far enough out you will find gaps as a large as you like, but the distribution of primes is very complicated and I agree it almost seems random but it really is not. Exactly what underlies the distribution is not understood but Reimann's Hypothesis is thought to strike at the heart of the question. That hypothesis sees primes as sport of intimately married to a certain complex function called the Reimann Zeta Function. I personally enjoy studying primes form the perspective of elementary number theory and Diophantine Equations.

I've never seen GH Hardy text but I would imagine it is hard. A good starter text that does not require much beyond precalc is Elementary Number Theory 3rd Ed. David M. Burton Hardy is famous for proofs in analytical number theory though, and was a mathematician of the highest caliber.

Yup, however I can have a dataset {1,2,4,5,2,3,1}, but this is tricky because it is actually {o1,o2,o3,o4,o5,o6,o7} seven distinct osbervations and not necessarily ordered either. For the purpose of data analysis I do not want to do this {1,2,4,5,2,3,1}={1,2,3,4,5}. The means are 9 and 7.5 respectively. So context can sometimes make or break the intepretation of symbols, thought it should not be. I suppose we could use square bracecs for datasets ot some other such thing like {d:1,2,4,5,2,3,1}. Context is usually sufficient. But I will say that if you have read hundreds of books on set theory over fifteen years and you have never seen the definition Matt mentioned, umm you might want to change the way you read math texts. EJ

Thats a deep question. Are you studying foundations? I had a great little book once by Paul Cohen where he proved that the continuum hypothesis is optional as an axiom. I loned it to an undergrad and never saw it again.

There are uses to number theory that happen in pure mathematics. For instance some algebraic proofs about the number of finite abelian groups of given order or something like this rely upon number theory proofs. Number theory does not gets its due respect these days but it still contains some of the most difficult problems anywhere, like Goldbach's Conjecture. Show that even natural number greater four can be written as the sum of two odd primes. Don't spend too much of your life trying, its been unsolved for well over 100 years. Trig is really the genesis of six special functions and a host of derived functions, but its so strongly linked to geometry and therefore engineering that I think its usefulness is obvious. And there are boat loads of intersting trig identities to. There is a very intersting structure to how those six basic trig functions combine under addition and multiplication.