mm-324 === Subject: : Re: Algebra shows algebraic integer limitation> For about two years now I've been trying to explain an algebraic> method for analyzing polynomials that relies on non-polynomial> factors.[snip]For an equally long time a very large audience has empiricallydemonstra by simple explicit evalution of your ravings that you area complete incompetent and a psychotic idiot.Hey stooopid loud troll , put up or shut up,http://www.rsasecurity.com/rsalabs/challenges/factoring/ faq.htmlhttp://www.rsasecurity.com/rsalabs/challenges/ factoring/numbers.htmlhttp://www.crank.net/harris.html It's not every braying jackass that gets a whole page at crank.netIs a $10,000 prize no questions asked too small to justify yoursubmission of two little prime numbers?-- Uncle Alhttp://www.mazepath.com/uncleal/ (Toxic URL! Unsafe for children and most mammals)Quis custodiet ipsos custodes? === Subject: : Re: JSH: Problem with Decker, Madigin, Ullrich et aluh-oh; he divided euronecrogeometers into two categories:those you hate to love, and the other kind. if there are some other (N) kind,don't sue me! well, the second decadeof the Ten Year Program in Marketing is going to be,really ... I just cannot say! > before you, as to them it was a refuge from the often crazy world of > Do you hate Gauss? Despise Dedekind? Spit on Archimedes? Feel> disgust for Newton? Blast Euler?--ils duces d'Enron!http://tarpley.net=== === Subject: : Re: JSH: Past posters, consider facts Okay, just so I get this...>A ring 'garantees' that if you add A and B, and A and B are on that>ring, the result A+B is also on that ring. A ring also 'garantees'>that A-B is on that ring (because every element has an additive>inverse), so substraction is covered too. Multiplication is also>'covered' (A*B is on the ring as well).It was my understanding that subtraction is not covered.1-2 is not a Natural Number.--=== === Subject: : Re: JSH: Past posters, consider facts > === Subject: : Re: JSH: Past posters, consider facts >Message-id: Okay, just so I get this... >A ring 'garantees' that if you add A and B, and A and B are on that >ring, the result A+B is also on that ring. A ring also 'garantees' >that A-B is on that ring (because every element has an additive >inverse), so substraction is covered too. Multiplication is also >'covered' (A*B is on the ring as well). > It was my understanding that subtraction is not covered. > 1-2 is not a Natural Number.It is covered. The Natural Numbers do not form a ring.You can find some basic terminology defined on: ( this shows also some difference in terminology between writers).Also note that I use the word properties on that page. There aretexts around that call them axioms. That is a serious misnaming.-- === === Subject: : Re: JSH: Past posters, consider facts>>A ring 'garantees' that if you add A and B, and A and B are on that>>ring, the result A+B is also on that ring. A ring also 'garantees'>>that A-B is on that ring (because every element has an additive>>inverse), so substraction is covered too. Multiplication is also>>'covered' (A*B is on the ring as well).>It was my understanding that subtraction is not covered.>1-2 is not a Natural Number.No, the natural numbers are not a ring. A ring has an additive identity(0) and additive inverses.Rings don't require a multiplicative inverse, and the usual definitiondoesn't require a multiplicative identity.-- Richard-- Spam filter: to mail me from a .com/.net site, put my surname in the headers.FreeBSD rules!=== === Subject: : Re: JSH: Past posters, consider A ring 'garantees' that if you add A and B, and A and B are on that>ring, the result A+B is also on that ring. A ring also 'garantees'>that A-B is on that ring (because every element has an additive>inverse), so substraction is covered too. Multiplication is also>'covered' (A*B is on the ring as well).>>It was my understanding that subtraction is not covered.>>1-2 is not a Natural Number.>No, the natural numbers are not a ring. A ring has an additive identity>(0) and additive inverses.Ok, my understanding is wrong, but subtraction is still not a ring operation,is it?>Rings don't require a multiplicative inverse, and the usual definition>doesn't require a multiplicative identity.>-- Richard>-- >Spam filter: to mail me from a .com/.net site, put my surname in the headers.>FreeBSD rules!--=== === Subject: : Theorem by David Gale re: geometric embeddability of neighborly polytopes?Hi all,Awhile back, a prof of mine proved for me essentially the followingfact:for any space of dimension d>=4, there is a neighborly polytope, withsuitably many vertices, that exists in a geometrically happy mannerin the space.The proof of this (surely poorly sta) fact was simple, slick, andmade use of something called the moment curve, which for R^n wouldbe the set of points of the form (t, t^2, ..., t^n) for t in R.I've had trouble finding information on this theorem.(a) Can somebody please state this theorem correctly/precisely?(b) Can someone prove it?In lieu of (a) and/(b), I'd be perfectly happy with a webreference....thanks for any assistance,cdj=== === Subject: : Re: Theorem by David Gale re: geometric embeddability of neighborly polytopes? 3QLpj-NoP*NzsIC,boYU]bQ]H'y<#4ga3$21:> for any space of dimension d>=4, there is a neighborly polytope, with> suitably many vertices, that exists in a geometrically happy manner> in the space.> The proof of this (surely poorly sta) fact was simple, slick, and> made use of something called the moment curve, which for R^n would> be the set of points of the form (t, t^2, ..., t^n) for t in R.> I've had trouble finding information on this theorem.One way of stating it is that any set of points on the moment curve form the vertices of a neighborly polytope. The combinatorial structure of the polytope depends only on the dimension and the number of points, and polytopes having this structure are called cyclic polytopes.If you do a Google search for cyclic polytope you will find a reasonable amount of web information.-- David Eppstein http://www.ics.uci.edu/~eppstein/Univ. of California, Irvine, School of Information & Computer Science=== === Subject: : Re: rotations w quaternions and eulersanother sort of representation that I've never done, butthe problem is still just to transform two rotations,taking either pair of the 3 inverse trig functions (the angles). if only one of the angles is a fixed (constant) one, thenits values can be anywhere on that circle around the axis,using the usual three orthogonal axes. the values of the other will be within a certain range (andthe 3rd axis's angles will be somehow complimentary to them). one can either use two of the angles with a sign operator,to dystinguish the octant of the unit-vector (say), orone can use three homogenous variables. one hemisphere of the sphere will map stereographicallyto the unit disk, and the other to the outside the disk (orto the other side of the coin .-) >>Maybe the problem lies in the fact that the euler angles describe a>>series of rotations.. and thus I perhaps miss something when converting>>from Euler to quaternion. > After that I extract Euler angles from the new quaternion. This seems to> work when my all angles are initially 0 and there is a single non-zero> angular velocity, but fails in other cases. Maybe the problem is that I am> concatenating all the angular velocites together instead of applying them> separately. Any ideas?--ils duces d'Enron!http://tarpley.net=== === Subject: : Re: Sum (A(k)*sin(kx) = Constant => How to determine A(k) by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i0MKmKU04015;To be more precise, I would say :Sum (A(k)*sin(k*pi*x/L) )=Cwhere x is not zero. L is a positive constant.C is another constant positive or negative. === === Subject: : Very Ugly Sum by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i0MN7nc16259;The following series converges for all t>0:1*exp(-t)+2*exp(-4t)+3*exp(-9t)+.....+n*exp(-n^2*t)+....I would like to know if it is possible to express it in a closedform,or to give integral expression for this series.Regards=== === Subject: : Re: Very Ugly Sum>The following series converges for all t>0:>1*exp(-t)+2*exp(-4t)+3*exp(-9t)+.....+n*exp(-n^2*t)+....> I would like to know if it is possible to express it in a closed>form,or to give integral expression for this series.I believe not, although something very close occurs in an integral transformrela to the Riemann Zeta function. See Titchmarsh (sp?) and some papers byVarga and Csordas in the late 80's=== === Subject: : Re: Elementary Inequality for Fun by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i0NDiVt14010;Hi!I think c is between 0 and x, but doesn't matter.Thx for the proof. However, here we must assume that by classical we mean to compute, e.g. 1.57*0.43*0.57^4.Also, I'm in very doubt that, for example, the following rela inequality: B-A>1/2 can be achieved in your manner.Ady.>> I'm asking for a classical (i.e., by hand, without using any calculators) >> solution of the following inequality:>> Let A=(e/2)^(e/2) and B=(pi/2)^(pi/2). Then AB>3.>By Taylor's theorem, for x > 0 we have> (1+x)^p = 1 + px + p(p-1)x^2/2! + p(p-1)(p-2)c^3/3!,>for some c between 1 and 1 + x. But if 1 < p < 2, the last term is negative >and we can say> (1+x)^p > 1 + px + p(p-1)x^2/2! + p(p-1)(p-2)x^3/3! (*)>Let's see: e/2 > (2.7)/2 = 1.35 and pi/2 > (3.14)/2 = 1.57. So> (e/2)^(e/2)*(pi/2)^(pi/2)> [(1+.35)^(1.35)]*[(1+.57)^(1.57)].>Now use (*) on each expression in brackets. After the annoying arithmetic >(which can be done classically), I got the last product > 3.04.=== === Subject: : Re: Connecness in the plane by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i0NDiUp13973;Gday -as mentioned elsewhere, B and D are path connecand hence connec.A has a partition U,V withU = (-inf, sqrt(2)) x RV = (sqrt(2), inf) x Rso it is disconnec. Similarly, C is disconnec.But to my knowledge, the best answer to your first questionis that, in manifolds (such as RxR) connec sets are path connec (and vice versa of course). I suppose this is a less specific characterisation than the intervals in R, isn't it.-dave.>Hi everybody,>on the real line we know exactly which sets are connec->the intervals.Is something similar true for the plane?>I mean,is there some characterization of the connec sets in RxR?>Even for relatively harmless sets in RxR such as>A={(x,y):x and y are rational}>B={(x,y):x or y is rational}>C={(x,y):x and y are irrational}>D={(x,y):x or y is irrational}>I don't know if they are connec.Which of them are?Why?>Any help welcomed.=== === Subject: : Re: Fundamental basics of unilaterality by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i0NDiUI13968;>>As an engineer, I do not need at all the fictitious notion of an> infinitely small value zero. Its neighbours 0- and 0+ do the same job.> Why not declaring the exact zero Cantors pipe dream? >>If you have 10 euros and then a mugger takes away those 10 euros>>from you, how many euros have you left? 0+? 0-? or just 0?>0+ if you were at least able to remember what he looked like>when going to the police. Otherwise 0-.Seriously, his questiond) Aren't 0+ and/or 0- quite sufficient? I think he means: do there exist a pair of positive real numbersthat are so insanely close to zero that we will never be able to measure physical quantities in macroscopic units that areeven closer to zero than these quantities? In other words, if you have 10 euros and the mugger takes everything but leaves youwith 10^{-23} euros, didnt he just go ahead and take everything? This question has been asked many, many times in different forms,all of which are basically all of the form: do physical quantities really not exist in the physical world just because they can't bemeasured (simultaneously or acurately or or ...)? As for the rest of his osophy (?), I didnt get it.Perhaps because of the use of all the buzz words. However, justbecause he uses all the buzz words in a totally incoherent mannerdoesnt automatically make him an idiot: he could also be implicitely assuming that if little he has read all these words at some point, then someone as high up as a professor ofmathematics... In other words, possibility one is that he is anidiot. Possibility two is that is, in his own way, giving complements, to which we add insults. >>Needless to say, I had the last laugh.>> Alan Partridge, _Bouncing Back_ (14 times)=== === Subject: : Solvable Polynomials of Degree 4m by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i0NDiYM14123;, Let P(x) be a particular solvable polynomial of degree 4m with aresolvent of degree 4m-1. What are the possible values of 4m? For degree 4 and some degree 8 (with transitive group T25), they can have a resolvent of degree 4m-1.degree 11 resolvent. What is the next degree then? 16, 20, etc?Sincerely,T. Piezas IIItpiezasQQQ@uap.edu.ph(Remove QQQ for real email)=== === Subject: : Quaternions and the determinant. by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i0NDiYh14133;Well, Ive searched the web and found a lot linking thequaternions with the vector cross-product. However, at least until now, I am yet to find linkage between quaternions anddeterminants. For example, from www.math.niu.edu/~rusin/known-math/96/octonionicI found****************************************************A general vector in 4-space (which we will call a quaternion)is given by the linear combination x = a . 1 + b . i + c . j + d . k or simply x = a + bi + cj + dk, where a, b, c, d are real numbers (the coordinates of the vector x in the given basis 1,i,j,k). The coefficient a is called its real part a = Re(x) and the remaining rest bi + cj + dk=Im(x) its imaginary part. A conjugation is defined by Conj(x) := Re(x) - Im(x) . The space of imaginary vectors is just ordinary 3dimansional space and we will see that the cross product is simply derived from the product on the quaternions as follows:For quaternions x and y define the cross product x X y as x X y = -1/2 Im( Conj(y).x ) .On the imaginary part this restricts to x X y = Im (x.y) and is just the ordinary cross product with e.g. the following properties: (1) x X y is alternating: y X x = - x X y ; (2) |x X y| = |x wedge y| . ************************************************************** ***My relations are:Let Q(A) be the quaternion algebra over the reals.For x = (x_1)i + (x_2)j + (x_3)k + (x_4)1, defineres: Q(A) -> R^3, res(x) = (x_1, x_2, x_3) andtes: Q(A) -> R , tes(x) = x_4Then for all x, y, z in Q(A) with x_4 = y_4 = z_4 = 0 the following holds:(i) res(x) scalar-product res(y) = - tes(x*y)(ia) ||res(x)|| = sqrt( - tes(x*y) ), euclidean norm (ii) res(x) cross-product res(y) = res(x*y)(iii) det( res(x), res(y), res(z) ) = -tes(x*y*z)(iv) res(x) cross-product (res(y) cross-product res(z)) = res(x*y*z)I will try to post a list of various proofs ofproperties, later. In the meantime, the reader can check.(see also mathforum.org/discuss/sci.math/t/572552 for some ofthese proofs. Note: the def. of determinant given in my firsttwo postings there was incorrect.) === === Subject: : Re: Action Device Tragedy by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i0NDiYr14129; >> 17 years ago, in 1986, when I was 17 year old, I ran away from one of>> the most prestigious engineering college in India, VJTI Bombay. Reason>> was, I don't want to be Engineer. I want to become Pilot. through which we can hang the things in air. We can build air vehicles>> and any one can become Pilot. But I am back to square 1. I will have to become Engineer to build>> this action device. There are no mechanical engineers in any workshop>> in this town. All you people in world are forcing me to build this>> device myself even though you are aware that at least one person has>> built one of the parts of this device and it works. Shame! Shame!>> Shame! I don't want to be Engineer. I want to be Pilot, you Morons! I am not going to build this action device. That spring in hardware>> shop struck to my finger very hard. -Abhi.[snip]>Kismat, Nasheeb, Destiny. I have comple one circle and reached to>same spot from where everything began. God is playing chess with me.>If it is so, my behaviour, moves will be same as it were 17 years ago.>I am not going to change it.>-Abhi.**************************************************** *****More than 1 billion indians and we got Abhi...**sigh**. What're theodds?And god doesn't play chess, he just isn't that smart. Actually he'spretty beatable in backgamoon...Tonio=== === Subject: : Re: HOW by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i0NEwRK20113;>HOW DO I GET A COPY OF THIS?Just place it on the glass and push copy.=== === Subject: : Re: Please read my preprint for a proof of Goldbach Conjecture by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i0NG7Fq25685;Would my argument be independent of, has nothing to do withyour remark? If independent, then why? This would be all I wouldbe asking for.As I said earlier, unless the questions above are notanswered properly, we must continue this postings.Thanks for your strictness, though.Hisanobu Shinya=== === Subject: : Mahlo Cardinals by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i0NJ6R708477;I read a definition of Mahlo Cardinals in Rudy Rucker's book (Infinity and the mind).It's probably not the best place for such definitions, and I didn't look at more serious books yet.Rudy gives what he calls a formal definition. He says that a Mahlo Cardinal is an inaccessibe cardinal which for every set of ordinals smaller than itself with it's cardinality, there is smaller inaccessible cardinal kappa, for which the set of all ordinals in that set less than kappa has a cardinality of kappa.I'd like to know if that's the way they usually define Mahlo Cardinals.The Levantine.=== === Subject: : areas of overlapping circles by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i0NJHH009361;>I would like to know a formula for calculating the area of>the overlapping region of two circles that overlap and touch the center of the other circle (knowing>just the radius).>Could someone please provide a formula, or point me towards>a useful reference?=== === Subject: : Re: Generating function of Repunits? by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i0OKRte18901;>I'm interes in what's the generating function of repunits?. That>is, what's the rational function q(x) such that>q(x) = r_1 + r_2x + r_3x^2 + ....>where r_n is the n-th repunit number.>Thank you very much,>Xan.I'm not sure I understand your question . . .You ask for the genrating funtion of repunits,then we are seeking a rational function q(x) ?My understanding is that a repunit is an integer composed of 1's such as: 1, 11, 111, 1111, etc.The n-th repunit would be: (10^n - 1)/9=== === Subject: : Re: Sum (A(k)*sin(kx) = Constant => How to determine A(k) by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i0OKRvF18994;The formula is supposed to be valid for an interval of x.How do we obtain : A(k) = (2/pi) * (1 - (-1)^k)) / k ?????What is the formula yielding the previous expression.Thanks very much>> I am dealing with the following problem :>> Sum of (A(k)*sin(kx) = B from k=0 to infinity>> where A(k) is a constant and B is a constant.>> Supposing I know the value of B,how do I find the coefficients>> A(k) as a function of B ??>> How is it rela to Fourier integrals.>> thanks in advance for your help>Something is missing from your question. As it stands, if we insert the>for every real x quantifier, then the only answer would be B=0 and>all A(k)=0. Why? Plug in x=0.> In more detail: The left side is an odd function, and the only constant>function which is an odd function is zero.> So, you have no freedom in specifying B.>Incidentally, A(0) need not be there since sin(0*x)=0.>So:>Is x just one prescribed number, different from an integer miltiple of pi?>Or: Is the formula supposed to be valid for an interval of values of x,>such as 0 < x < pi (and extends as an odd function)?> If so, any (honest) textbook that presents Fourier Series has the>answer:> A(k) = (2/pi) * (1 - (-1)^k)) / k>Or: What is it you really want?> ZVK(Slavek).=== === Subject: : Re: Almost Sure Convergence To Zero by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i0OKRuu18970;>>I'm taking a course in probability theory and I'm trying>>to prove the Strong Law of Large Numbers without using >>Kolmogorov inequality. >>X1,X2,... are independent random variables,with expec value 0>>such that the series Var(Xn)/nn converges.>>I can prove that Xn/n --->0 almost sure.>>Does it follow from this that (X1+X2+....+Xn)/n --->0 almost sure?>No. For example, if X_n are uniformly bounded then certainly >X_n/n -> 0 (not just almost surely, and without any requirement >on the expec values), but (X_1+X_2+...+X_n)/n need not go to 0.>Robert Israel israel@math.ubc.ca>Department of Mathematics Thank you!I didn't expect to give a new proof,but at least Itried.So,Kolmogorov is *THE LAW* in the Law of Strong Numbers.It is really interesting to ask if there is any other proofnot using this fundamental inequality.Edward === === Subject: : Re: Squaring A Circle...(almost) by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i0OKRuM18977;>>I'm sorry, but you never make it clear what you mean by square a>>circle. Please explain.>As for squaring the circle, I believe this phrase is odd, actually.>(I believe I have also heard rectifying the circle, as well.)>So, what are you doing when you take a compass, put its point in the>middle of a square, and use it to draw a circle around the square?...>Circling the Square, of course!...May be the following sheds light:http://www.stefanides.gr/piquad.htmPanagiotis Stefanideshttp://www.stefanides.>Leroy Quet=== === Subject: : System Linear Transformation / Common Info. Properties by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i0OKRtE18909;Hello Math. Forum:I am stumped. I amy trying to come up with a System wide Linear Transformation expressing common informational properties. This would be used in Matrix Algebra. It would be relational and non axiomatic. I think.What is a common informational property? An equality expressing the relationship between bits and software and output of applications , would be one example. I want an equality that would express this relationship for other systems as well.Zim Olson, http://www.zimmathematics.comVisiting Faribault, MN Library=== === Subject: : Re: Quaternions and the determinant. by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i0P1SnR07503;>Well, Ive searched the web and found a lot linking the>quaternions with the vector cross-product. However, at least >until now, I am yet to find linkage between quaternions and>determinants. For example, from >www.math.niu.edu/~rusin/known-math/96/octonionic>I found>****************************************************>A general vector in 4-space (which we will call a quaternion)>is given by the linear combination > x = a . 1 + b . i + c . j + d . k >or simply > x = a + bi + cj + dk, >where a, b, c, d are real numbers (the coordinates of the vector x >in the given basis 1,i,j,k). >The coefficient a is called its real part > a = Re(x) >and the remaining rest > bi + cj + dk=Im(x) >its imaginary part. >A conjugation is defined by > Conj(x) := Re(x) - Im(x) . >The space of imaginary vectors is just ordinary 3dimansional space >and we will see that the cross product is simply derived from the >product on the quaternions as follows:>For quaternions x and y define the cross product x X y as > x X y = -1/2 Im( Conj(y).x ) .>On the imaginary part this restricts to > x X y = Im (x.y) >and is just the ordinary cross product >with e.g. the following properties: > (1) x X y is alternating: y X x = - x X y ; > (2) |x X y| = |x wedge y| . >************************************************************* ****>My relations are:>Let Q(A) be the quaternion algebra over the reals.>For x = (x_1)i + (x_2)j + (x_3)k + (x_4)1, define>res: Q(A) -> R^3, res(x) = (x_1, x_2, x_3) and>tes: Q(A) -> R , tes(x) = x_4>Then for all x, y, z in Q(A) with x_4 = y_4 = z_4 = 0 the >following holds:>(i) res(x) scalar-product res(y) = - tes(x*y)>(ia) ||res(x)|| = sqrt( - tes(x*y) ), euclidean norm >(ii) res(x) cross-product res(y) = res(x*y)>(iii) det( res(x), res(y), res(z) ) = -tes(x*y*z)>(iv) res(x) cross-product (res(y) cross-product res(z)) = res(x*y*z)>I will try to post a list of various proofs of>properties, later. In the meantime, the reader can check.>(see also mathforum.org/discuss/sci.math/t/572552 for some of>these proofs. Note: the def. of determinant given in my first>two postings there was incorrect.)(v) res(x) scalar-product (res(y) cross-product res(y)) = = det( res(x), res(y), res(z) )proof of (v): res(x) scalar-product (res(y) cross-product res(z)) = = res(x) scalar-product res(y*z), from (ii) = - tes( x*(y*z) ), from (i) = - tes( x*y*z ) = det( res(x), res(y), res(z) ), from (iii)q.e.d.Just a check: Let I be the (3 by 3) identity matrix, then det(I) = det( res(i), res(j), res(k) ) = = - tes( i*j*k ) = - tes( -1 ) = 1 === === Subject: : Re: Fibonacci PHI, PI, e Relation by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i0OKRv619003;Last one.This list does 1,000,000 decimal digits --- There are 9997 numbers in the list.{12, 99, 169, 395, 499, 595, 606, 693, 824, 827, 840, 940, 1282, 1291, 1384, 1594, 1705, 1742, 1905, 2020, 2060, 2153, 2257, 2302, 2359, 2367, 2507, 2546, 2557, 2710, 2724, 2791, 2832, 2857, 3036, 3051, 3280, 3309, 3429, 3497, 3518, 3591, 3651, 3709, 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259436, 259553, 259576, 259773, 259877, 259889, 259956, 260030, 260141, 260196, 260203, 260514, 260616, 260681, 260682, 260713, 260730, 260746, 260834, 260917, 260964, 261014, 261099, 261103, 261147, 261194, 261443, 261449, 261456, 261485, 261627, 261649, 261767, 261798, 261841, 261854, 261875, 261904, !261977, 262067, 262228, 262673, 263080, 263201, 263254, 263256, 263272, 263297, 263359, 263488, 263588, 264223, 264510, 264744, 265240, 265287, 265323, 265419, 265451, 265591, 265641, 265664, 265826, 265849, 265915, 265940, 265951, 266050, 266052, 266243, 266447, 266503, 266520, 266684, 266781, 267060, 267212, 267248, 267317, 267472, 267488, 267518, 267564, 267637, 267674, 267769, 267782, 267852, 267899, 267914, 267966, 268018, 268107, 268165, 268409, 268690, 268748, 268758, 268964, 268978, 269033, 269749, 269783, 269784, 269969, 270049, 270146, 270449, 270725, 270825, 270841, 270895, 270935, 271265, 271361, 271407, 271451, 271482, 271499, 271554, 271712, 271797, 271901, 271911, 272017, 272172, 272247, 272501, 272593, 272633, 273232, 273246, 273256, 273414, 273470, 273628, 273653, 273742, 273778, 273845, 273881, 273887, 274025, 274133, 274149, 274323, 274397, 274493, 274504, 274594, 274632, 274743, 274857, 274909, 274928, 275004, 275046, 275250, 275553, 275563,! 275592, 275687, 275802, 275873, 275978, 276434, 276543, 276562, 276574, 276613, 276989, 277174, 277203, 277356, 277365, 277366, 277600, 277604, 277738, 277783, 278043, 278192, 278596, 278632, 278721, 279012, 279389, 279779, 279799, 279871, 279875, 279884, 279912, 280063, 280075, 280117, 280213, 280373, 280396, 280410, 280515, 280534, 280762, 280785, 280896, 280900, 280938, 280983, 281281, 281287, 281331, 281414, 281822, 281882, 281901, 282044, 282182, 282509, 282531, 282553, 282635, 282727, 282734, 282824, 283004, 283210, 283424, 283459, 283482, 283668, 283705, 283815, 283942, 283984, 284045, 284146, 284185, 284241, 284340, 284409, 284654, 284777, 284956, 284972, 285005, 285011, 285107, 285259, 285304, 285313, 285504, 285512, 285567, 285933, 286127, 286387, 286556, 286582, 286743, 286767, 286784, 286895, 286942, 287045, 287287, 287332, 287357, 287631, 287722, 287921, 287997, 288012, 288102, 288109, 288440, 288516, 288712, 288806, 288829, 288931, 289331, 289448,! 289637, 289688, 289761, 290025, 290061, 290067, 290070, 290140, 290336, 290469, 290529, 290599, 290978, 291150, 291210, 291300, 291415, 291498, 291532, 291585, 291593, 291882, 292101, 292160, 292246, 292279, 292282, 292288, 292445, 292730, 292764, 292988, 293116, 293163, 293174, 293290, 293362, 293445, 293593, 293643, 293665, 293698, 293830, 293832, 293999, 294015, 294243, 294388, 294412, 294473, 294676, 294746, 294749, 294992, 295200, 295240, 295283, 295316, 295341, 295484, 295493, 295554, 295573, 295672, 295676, 295721, 295812, 296022, 296085, 296121, 296148, 296189, 296211, 296225, 296322, 296334, 296394, 296425, 296581, 296613, 296702, 296830, 296863, 297074, 297402, 297566, 297781, 297815, 297956, 297995, 298195, 298205, 298236, 298253, 298288, 298299, 298541, 298593, 298692, 298720, 298822, 298831, 298875, 299022, 299049, 299136, 299242, 299296, 299314, 299538, 299606, 299620, 299663, 299693, 299707, 299840, 299861, 300005, 300052, 300138, 300257, 300275!, 300356, 300464, 300576, 300584, 300731, 300870, 300938, 301049, 301209, 301221, 301533, 301536, 301588, 301720, 301772, 301776, 301898, 301909, 301993, 302066, 302114, 302429, 302463, 302468, 302571, 302714, 302844, 302902, 303003, 303017, 303137, 303176, 303208, 303242, 303263, 303300, 303373, 303510, 303556, 303568, 303574, 303935, 304040, 304196, 304224, 304404, 304411, 304504, 304660, 304751, 304788, 304887, 305053, 305071, 305133, 305143, 305447, 305510, 305679, 305710, 305952, 306128, 306241, 306278, 306366, 306517, 306630, 306727, 306837, 307088, 307311, 307444, 307593, 307668, 307693, 307826, 307855, 307881, 307916, 307983, 308100, 308137, 308314, 308322, 308433, 308483, 308521, 308569, 308737, 308744, 308884, 309258, 309278, 309293, 309311, 309338, 309503, 309505, 309565, 309599, 309803, 309835, 309917, 309943, 310062, 310161, 310223, 310258, 310361, 310607, 311200, 311272, 311398, 311419, 311514, 311539, 311802, 311955, 312022, 312061, 312099, 312317!, 312396, 312481, 312677, 312702, 312732, 312806, 313084, 313364, 313366, 313866, 313870, 313995, 314034, 314044, 314081, 314138, 314181, 314230, 314257, 314315, 314447, 314519, 314573, 314660, 314747, 314850, 314949, 315118, 315146, 315382, 315438, 315477, 315557, 315719, 315831, 315920, 316001, 316205, 316225, 316308, 316660, 316765, 316801, 316968, 316973, 317051, 317055, 317398, 317495, 317524, 317570, 317650, 317749, 317858, 317919, 317944, 318511, 318517, 318534, 318566, 318596, 318648, 318743, 319040, 319226, 319280, 319663, 319722, 320230, 320236, 320257, 320298, 320429, 320925, 320932, 320953, 321049, 321384, 321391, 321492, 321606, 321675, 321897, 321912, 322179, 322188, 322208, 322366, 322414, 322422, 322634, 322770, 322872, 322976, 323071, 323183, 323327, 323402, 323422, 323525, 323604, 323616, 323650, 323840, 323851, 323916, 323930, 324475, 324614, 324849, 325019, 325225, 325358, 325361, 325408, 325471, 325496, 325501, 325569, 325637, 325776, 32592!0, 325981, 325984, 326178, 326185, 326276, 326554, 326727, 326858, 326912, 326987, 327209, 327234, 327256, 327259, 327262, 327343, 327402, 327439, 327785, 327926, 328063, 328075, 328503, 328951, 329159, 329371, 329392, 329825, 329862, 329866, 330005, 330173, 330327, 330359, 330397, 330411, 330471, 330523, 330527, 330594, 330683, 330693, 330770, 330828, 330875, 330965, 331195, 331285, 331319, 331396, 331827, 331846, 331880, 331941, 332017, 332057, 332169, 332187, 332259, 332273, 332329, 332520, 332658, 332700, 332914, 332925, 332950, 333074, 333091, 333176, 333192, 333318, 333385, 333426, 333530, 333543, 333616, 333664, 333774, 333790, 333836, 333877, 334158, 334293, 334335, 334337, 334423, 334448, 334534, 334603, 334743, 334761, 334862, 335094, 335234, 335371, 335615, 335635, 335729, 335751, 335858, 336019, 336020, 336037, 336123, 336197, 336223, 336449, 336484, 336488, 336523, 336623, 336757, 336837, 336922, 336987, 337127, 337328, 337375, 337429, 337640, 33777!6, 338012, 338101, 338149, 338156, 338301, 338347, 338387, 338446, 338469, 338854, 338856, 338895, 338926, 338997, 339019, 339120, 339121, 339175, 339280, 339394, 339463, 339509, 339592, 339694, 339742, 339803, 339914, 339992, 339998, 340022, 340043, 340190, 340240, 340270, 340295, 340621, 340716, 340717, 340740, 340823, 340986, 341082, 341092, 341115, 341157, 341296, 341575, 341607, 341751, 341999, 342028, 342059, 342143, 342284, 342337, 342486, 342618, 342761, 342776, 342789, 342953, 342992, 343096, 343109, 343222, 343360, 343422, 343468, 343477, 343592, 343672, 343721, 343780, 343867, 343903, 343971, 344047, 344108, 344138, 344375, 344402, 344885, 344934, 344970, 345130, 345321, 345420, 345478, 345583, 345604, 345637, 345658, 345731, 345868, 345883, 346096, 346184, 346193, 346304, 346373, 346412, 346574, 346611, 346658, 347017, 347170, 347241, 347266, 347582, 347595, 347682, 347747, 347794, 347803, 347876, 348012, 348244, 348266, 348293, 348369, 348427, 3485!02, 348590, 348815, 349012, 349023, 349029, 349308, 349540, 349701, 349741, 349864, 350003, 350084, 350243, 350368, 350466, 350536, 350624, 350640, 350654, 350661, 350788, 350870, 350909, 351032, 351075, 351348, 351368, 351387, 351485, 351573, 351655, 351963, 352048, 352290, 352432, 352689, 352929, 353273, 353296, 353594, 353710, 353784, 353952, 353963, 354271, 354345, 354356, 354476, 354493, 354522, 354605, 354655, 354982, 355075, 355104, 355174, 355175, 355242, 355273, 355319, 355442, 355460, 355648, 355764, 356005, 356072, 356582, 356906, 356970, 357189, 357268, 357282, 357346, 357352, 357615, 357662, 357689, 357714, 357999, 358005, 358093, 358109, 358182, 358186, 358195, 358217, 358249, 358500, 358727, 358781, 358829, 358969, 359020, 359119, 359125, 359204, 359239, 359251, 359299, 359324, 359406, 359419, 359591, 359604, 359665, 359840, 359884, 360119, 360315, 360501, 360819, 360901, 360925, 360996, 361191, 361204, 361260, 361290, 361473, 361539, 361551, 361!045, 372183, 372199, 372263, 372445, 372461, 372524, 372535, 372676, 372898, 372923, 372940, 373013, 373035, 373052, 373101, 373184, 373553, 373849, 373986, 374012, 374063, 374223, 374256, 374558, 374566, 374707, 374779, 374939, 374941, 375157, 375252, 375427, 375462, 375486, 375559, 375580, 375828, 375861, 375932, 376048, 376056, 376166, 376313, 376458, 376670, 376784, 376806, 376881, 376882, 376934, 377308, 377542, 377593, 377788, 377940, 378041, 378101, 378134, 378196, 378238, 378469, 378694, 378917, 379197, 379233, 379236, 379369, 379523, 379535, 379847, 379930, 380301, 380471, 380551, 380751, 380823, 380935, 381050, 381329, 381343, 381370, 381424, 381470, 381673, 381678, 381830, 381913, 382193, 382244, 382323, 382361, 382403, 382501, 382630, 382668, 382722, 382843, 382884, 383011, 383097, 383110, 383163, 383180, 383252, 383464, 383466, 383473, 383658, 383694, 384056, 384075, 384151, 384199, 384216, 384229, 384416, 384539, 384548, 384638, 384743, 384782, 38!4840, 384955, 385008, 385087, 385119, 385455, 385536, 385542, 385659, 385698, 385990, 386015, 386126, 386230, 386268, 386290, 386551, 386575, 386694, 386837, 386936, 386953, 387070, 387150, 387186, 387525, 387526, 387572, 387915, 388010, 388137, 388148, 388213, 388221, 388278, 388347, 388428, 388473, 388484, 388612, 388674, 388909, 388969, 389025, 389158, 389280, 389345, 389369, 389377, 389442, 389595, 389832, 389863, 389912, 390000, 390085, 390230, 390276, 390343, 390449, 390519, 390615, 390862, 390971, 391025, 391225, 391301, 391325, 391356, 391728, 391752, 391766, 391814, 392037, 392184, 392299, 392391, 392453, 392654, 393046, 393059, 393107, 393171, 393239, 393384, 393496, 393655, 393715, 393725, 393744, 393746, 393784, 393815, 393932, 394188, 394373, 394508, 394586, 394693, 394797, 394850, 394859, 394931, 395086, 395139, 395376, 395614, 395695, 395790, 396010, 396013, 396047, 396078, 396406, 396517, 396663, 396809, 396851, 396943, 396995, 397216, 397228, 39!7356, 397412, 397426, 397745, 397909, 398154, 398160, 398338, 398383, 398422, 398474, 398479, 398599, 398637, 398690, 398713, 398757, 398815, 399126, 399219, 399231, 399294, 399316, 399478, 399526, 399768, 399837, 399926, 400169, 400317, 400398, 400589, 400732, 400940, 400952, 401069, 401169, 401400, 401571, 401607, 401740, 401747, 401977, 402064, 402163, 402218, 402266, 402290, 402494, 402618, 402644, 402691, 402744, 402981, 403042, 403216, 403247, 403271, 403491, 403524, 403599, 403630, 403662, 403788, 403811, 403931, 404087, 404294, 404526, 404531, 404658, 404680, 404872, 404949, 404997, 405000, 405231, 405272, 405604, 405686, 405860, 405926, 406053, 406121, 406218, 406332, 406401, 406623, 406768, 406937, 406969, 406980, 407006, 407070, 407187, 407192, 407286, 407325, 407344, 407355, 407611, 407791, 407832, 407856, 407965, 408122, 408176, 408177, 408236, 408322, 408532, 408630, 408637, 408713, 408728, 408756, 408764, 408916, 408920, 408955, 408959, 409062, 4!09087, 409440, 409526, 410022, 410060, 410130, 410227, 410245, 410378, 410498, 410606, 410667, 410691, 410725, 410894, 411071, 411149, 411179, 411186, 411319, 411381, 411496, 411523, 411681, 411887, 412155, 412398, 412509, 412632, 412846, 412907, 412917, 413079, 413117, 413155, 413176, 413335, 413387, 413441, 413547, 413666, 413670, 413783, 413896, 413922, 413968, 414144, 414270, 414303, 414474, 414635, 414676, 414743, 414797, 414854, 414910, 415082, 415313, 415396, 415413, 415472, 415542, 415602, 415708, 415873, 415958, 416043, 416155, 416216, 416305, 416373, 416568, 416677, 416681, 416732, 416765, 416902, 416991, 417059, 417064, 417405, 417618, 417641, 417677, 417746, 418005, 418054, 418095, 418343, 418485, 418503, 418531, 418556, 418572, 418701, 418702, 418718, 418719, 418824, 419083, 419084, 419119, 419156, 419184, 419216, 419265, 419331, 419341, 419436, 419469, 419485, 419486, 419536, 419563, 419619, 419679, 419682, 419698, 419742, 419997, 420018, 420085, 4!20232, 420235, 420470, 420507, 420589, 420691, 421144, 421339, 421456, 421485, 421498, 421699, 421854, 421974, 421988, 422042, 422045, 422194, 422412, 422472, 422551, 422828, 422917, 422995, 423053, 423405, 423503, 423535, 423706, 423760, 423892, 424057, 424089, 424128, 424154, 424220, 424318, 424395, 424459, 424494, 424513, 424529, 424675, 424711, 425067, 425131, 425148, 425304, 425363, 425620, 425820, 425920, 426292, 426464, 426490, 426594, 426646, 426653, 426735, 426889, 426930, 426936, 427088, 427133, 427230, 427352, 427517, 427849, 427930, 427972, 428021, 428026, 428039, 428168, 428189, 428219, 428230, 428250, 428270, 428322, 428324, 428328, 428711, 428748, 428910, 429030, 429058, 429167, 429170, 429183, 429434, 429841, 429915, 429925, 430168, 430206, 430232, 430346, 430524, 430605, 430625, 430818, 431098, 431185, 431230, 431254, 431408, 431649, 431708, 431767, 431846, 431931, 431961, 432309, 432377, 432421, 432544, 432561, 432688, 432741, 432872, 432926, !432937, 432997, 433034, 433297, 433434, 433718, 433800, 433903, 433913, 434062, 434081, 434082, 434260, 434614, 434632, 434754, 434819, 434875, 434891, 434938, 435095, 435103, 435139, 435194, 435244, 435292, 435378, 435443, 435604, 435649, 435750, 436166, 436207, 436278, 436367, 436368, 436372, 436543, 436562, 436594, 436678, 436682, 436893, 436983, 437009, 437029, 437114, 437206, 437229, 437281, 437387, 437569, 437606, 437665, 437744, 437758, 437980, 438085, 438122, 438232, 438442, 438677, 438707, 438744, 438890, 438905, 438979, 439066, 439119, 439233, 439258, 439319, 439498, 439691, 439797, 440403, 441099, 441175, 441264, 441272, 441312, 441353, 441407, 441561, 441767, 442048, 442111, 442141, 442160, 442353, 442374, 442390, 442407, 442416, 442418, 442423, 442572, 443084, 443097, 443224, 443435, 443444, 443554, 443557, 443656, 443855, 443875, 443896, 443945, 443996, 444310, 444353, 444469, 444626, 445040, 445164, 445179, 445187, 445303, 445329, 445352, 445501, !445540, 445541, 445554, 445600, 445620, 445797, 445839, 445976, 446202, 446359, 446517, 446622, 446700, 446709, 446872, 446975, 446988, 447025, 447082, 447097, 447361, 447478, 447579, 447744, 447780, 448134, 448182, 448215, 448277, 448302, 448367, 448584, 448614, 448712, 448858, 449017, 449090, 449198, 449381, 449600, 449738, 449750, 449880, 449909, 449921, 450038, 450083, 450142, 450183, 450198, 450353, 450666, 450813, 450892, 450893, 451070, 451157, 451182, 451431, 451592, 451674, 451797, 451859, 451958, 452208, 452209, 452335, 452390, 452410, 452526, 452563, 452681, 452708, 452739, 452802, 452894, 453025, 453073, 453219, 453372, 453454, 453482, 453488, 453528, 453592, 453849, 453880, 453974, 454181, 454343, 454379, 454384, 454423, 454552, 454626, 454661, 454751, 454784, 454908, 455022, 455049, 455123, 455138, 455254, 455267, 455441, 455443, 455468, 455604, 455613, 455663, 456041, 456262, 456357, 456369, 456375, 456430, 456459, 456574, 456709, 456791, 456808,! 456843, 456857, 456983, 457068, 457123, 457321, 457618, 457767, 457829, 457931, 458057, 458296, 458553, 458755, 458863, 458993, 458998, 459343, 459397, 459450, 459647, 459750, 459754, 459825, 459849, 459894, 459942, 459947, 460093, 460104, 460292, 460334, 460412, 460642, 460686, 461020, 461151, 461164, 461297, 461350, 461390, 461495, 461540, 461693, 461738, 461801, 461839, 461864, 461866, 461941, 461975, 462072, 462132, 462133, 462155, 462350, 462375, 462462, 462548, 462898, 462921, 462988, 463165, 463339, 463413, 463416, 463446, 463473, 463532, 463574, 463769, 463959, 464062, 464088, 464093, 464245, 464271, 464277, 464291, 464367, 464429, 464517, 464797, 464851, 464873, 464880, 464948, 465153, 465155, 465199, 465247, 465282, 465448, 465768, 465823, 465957, 466064, 466124, 466341, 466410, 466412, 466599, 466610, 466617, 466651, 466663, 466734, 466738, 466884, 466887, 467019, 467146, 467213, 467450, 467599, 467646, 467741, 467864, 467900, 468028, 468074, 468084,! 468138, 468153, 468458, 468501, 468559, 468565, 468688, 468740, 469001, 469489, 469699, 469752, 469959, 469966, 469989, 470150, 470218, 470314, 470347, 470379, 470383, 470624, 470668, 470753, 470757, 470788, 470814, 471100, 471165, 471172, 471270, 471436, 471437, 471467, 471470, 471525, 471537, 471659, 471732, 471820, 471935, 472050, 472064, 472217, 472315, 472420, 472546, 472689, 472691, 472790, 472856, 473135, 473185, 473906, 473949, 473998, 474007, 474023, 474080, 474131, 474161, 474209, 474475, 474557, 474749, 474854, 474873, 474896, 475038, 475092, 475095, 475275, 475317, 475462, 475540, 475668, 475683, 475865, 475909, 476028, 476091, 476112, 476149, 476217, 476245, 476279, 476296, 476386, 476510, 476677, 476812, 476825, 476854, 477192, 477247, 477277, 477291, 477293, 477320, 477498, 477522, 477602, 477620, 477844, 477907, 478045, 478214, 478226, 478246, 478374, 478439, 478520, 478524, 478592, 478764, 478798, 478835, 479058, 479116, 479242, 479269, 479279!, 479387, 479483, 479693, 479700, 479771, 479855, 479967, 480054, 480187, 480612, 480629, 480735, 480860, 481285, 481410, 481470, 481471, 481526, 481640, 481769, 481795, 481838, 481856, 481937, 482080, 482104, 482118, 482662, 482667, 482682, 482827, 483000, 483019, 483124, 483131, 483269, 483378, 483388, 483431, 483452, 483486, 483622, 483765, 483894, 484063, 484241, 484310, 484392, 484399, 484755, 484779, 484889, 485132, 485206, 485317, 485434, 485481, 485568, 485753, 485964, 486061, 486136, 486351, 486390, 486493, 486539, 486557, 486594, 486609, 486741, 486743, 486990, 487171, 487176, 487225, 487493, 487520, 487556, 487592, 487821, 487868, 487914, 488078, 488329, 488379, 488440, 488468, 489193, 489399, 489536, 489703, 489763, 489834, 489901, 489929, 490037, 490172, 490225, 490335, 490398, 490404, 490433, 490520, 490563, 490601, 490713, 490728, 490762, 490791, 490803, 490841, 490900, 490941, 491061, 491179, 491223, 491397, 491654, 491712, 491849, 492158, 49230!7, 492570, 492594, 492877, 492983, 493040, 493077, 493320, 493340, 493480, 493533, 493570, 493571, 493652, 493661, 493740, 493761, 493762, 493814, 494044, 494149, 494337, 494395, 494477, 494627, 494652, 494671, 494755, 494768, 494835, 494906, 494958, 495000, 495203, 495324, 495332, 495400, 495410, 495586, 495597, 495751, 495897, 495947, 496038, 496187, 496207, 496312, 496411, 496421, 496497, 496566, 496859, 496883, 496904, 497023, 497099, 497265, 497387, 497406, 497464, 497483, 497539, 497687, 497731, 498005, 498046, 498116, 498318, 498326, 498442, 498470, 498644, 498670, 498841, 499060, 499169, 499320, 499459, 499554, 499594, 499749, 499851, 500090, 500278, 500394, 500439, 500459, 500503, 500749, 500771, 500792, 500890, 500944, 500970, 501000, 501297, 501333, 501460, 501538, 501654, 501657, 501692, 501768, 501812, 501852, 501930, 501995, 502230, 502607, 503142, 503259, 503268, 503272, 503315, 503360, 503430, 503468, 503525, 503537, 503584, 503586, 503663, 50373!7, 503809, 503839, 503847, 503899, 504032, 504079, 504264, 504477, 504490, 504639, 504696, 504869, 504880, 504909, 504918, 504922, 505148, 505223, 505250, 505452, 505533, 505589, 505641, 505727, 505733, 505766, 505979, 506181, 506344, 506359, 506399, 506599, 506655, 506670, 506748, 506815, 506821, 506965, 506973, 506980, 507052, 507072, 507183, 507231, 507249, 507424, 507530, 507876, 507889, 507911, 508039, 508080, 508395, 508554, 508628, 508763, 508978, 508993, 509022, 509288, 509297, 509370, 509375, 509491, 509554, 509615, 509729, 509803, 510106, 510107, 510229, 510416, 510456, 510748, 510796, 510912, 511015, 511107, 511120, 511129, 511266, 511532, 511554, 511956, 512160, 512213, 512343, 512350, 512478, 512554, 512585, 512624, 512729, 512966, 513032, 513067, 513089, 513133, 513396, 513402, 513427, 513515, 513624, 513626, 513796, 513812, 513996, 514038, 514099, 514388, 514466, 514576, 514644, 514714, 514782, 514805, 514948, 515010, 515139, 515141, 515146, 5153!05, 515522, 515576, 515646, 515659, 515802, 515981, 516010, 516025, 516120, 516126, 516190, 516302, 516393, 516542, 516581, 516694, 516834, 516846, 517085, 517126, 517222, 517298, 517528, 517634, 517648, 517678, 517863, 518151, 518155, 518181, 518245, 518256, 518279, 518286, 518403, 518499, 518714, 518728, 518753, 518822, 518926, 518943, 518962, 519099, 519284, 519298, 519310, 519391, 519503, 519572, 519582, 519585, 519634, 519670, 519823, 519904, 520128, 520135, 520171, 520229, 520294, 520562, 520656, 520788, 520794, 520874, 520883, 521106, 521249, 521281, 521293, 521383, 521491, 521511, 521648, 521696, 521798, 521868, 521950, 522107, 522217, 522258, 522554, 522627, 522890, 523042, 523102, 523434, 523451, 523621, 523723, 523791, 523795, 523900, 523911, 523931, 524013, 524073, 524147, 524182, 524304, 524584, 524805, 524845, 525028, 525092, 525327, 525358, 525388, 525443, 525480, 525612, 525620, 525765, 525768, 525932, 526012, 526134, 526209, 526310, 526419, 5266!71, 526752, 526854, 526878, 526883, 526891, 526906, 526959, 527073, 527330, 527447, 527533, 527564, 527579, 527605, 527635, 527903, 528071, 528211, 528320, 528508, 528587, 528602, 528645, 528702, 528878, 528917, 529000, 529087, 529106, 529195, 529436, 529508, 529617, 529651, 529682, 529923, 530047, 530097, 530130, 530510, 530550, 530627, 530673, 530843, 530884, 530892, 531182, 531350, 531365, 531373, 531528, 531546, 531580, 531639, 531641, 531768, 531921, 531933, 531979, 532001, 532031, 532095, 532240, 532282, 532400, 532574, 532732, 532836, 532949, 532951, 532996, 533103, 533145, 533251, 533261, 533374, 533395, 533431, 533466, 533513, 533687, 533882, 533961, 533995, 534070, 534132, 534189, 534192, 534287, 534291, 534399, 534467, 534667, 534788, 534850, 534852, 535093, 535105, 535112, 535277, 535311, 535459, 535467, 535745, 535762, 535849, 535877, 536093, 536110, 536231, 536270, 536421, 536534, 536626, 536867, 536969, 536989, 537029, 537047, 537104, 537143, 537!168, 537242, 537382, 537426, 537481, 537654, 537696, 537763, 537769, 537961, 537990, 538007, 538027, 538035, 538172, 538210, 538282, 538365, 538370, 538401, 538716, 538777, 538905, 539059, 539183, 539369, 539425, 539523, 539532, 539548, 539631, 539712, 539830, 539928, 539976, 539997, 540012, 540174, 540430, 541164, 541451, 541567, 541589, 541670, 541886, 542011, 542235, 542293, 542516, 542621, 542740, 543430, 543461, 543580, 543708, 543992, 544008, 544016, 544179, 544253, 544279, 544423, 544466, 544488, 544528, 544599, 544672, 544785, 544980, 545024, 545527, 545792, 545810, 545831, 545835, 545887, 545935, 546004, 546006, 546035, 546046, 546163, 546217, 546237, 546238, 546273, 546497, 546633, 546716, 546796, 546805, 546886, 547149, 547244, 547363, 547394, 547608, 547802, 547851, 548139, 548140, 548172, 548245, 548417, 548481, 548584, 548729, 548940, 549154, 549361, 549491, 549594, 549675, 549754, 549837, 550104, 550206, 550304, 550384, 550468, 550529, 550644, 550!703, 550746, 551039, 551057, 551096, 551279, 551321, 551324, 551350, 551395, 551453, 551867, 552043, 552120, 552180, 552233, 552273, 552463, 552482, 552542, 552672, 552861, 553090, 553155, 553205, 553307, 553337, 553387, 553523, 553593, 553644, 553721, 553802, 553843, 554031, 554151, 554270, 554626, 554724, 554769, 554813, 554871, 554950, 555198, 555258, 555305, 555381, 555538, 556299, 556311, 556419, 556510, 556599, 556729, 556793, 556913, 556921, 556940, 557017, 557073, 557288, 557569, 557653, 558174, 558219, 558378, 558611, 558675, 558782, 558933, 559017, 559021, 559107, 559170, 559184, 559421, 559486, 559596, 559680, 559761, 559763, 559863, 559967, 560506, 560732, 560980, 561290, 561398, 561409, 561619, 561895, 561985, 562058, 562071, 562243, 562263, 562282, 562289, 562611, 562649, 562676, 562856, 562895, 563095, 563384, 563413, 563446, 563538, 563613, 563621, 563626, 563683, 563728, 563744, 563772, 563794, 563848, 564097, 564099, 564217, 564772, 564970, 56!4989, 565030, 565092, 565230, 566012, 566188, 566251, 566301, 566316, 566600, 566615, 566698, 566723, 566895, 567077, 567092, 567098, 567131, 567143, 567210, 567257, 567658, 567706, 567878, 567939, 568219, 568252, 568298, 568370, 568381, 568426, 568553, 568665, 569136, 569370, 569396, 569407, 569435, 569511, 569747, 569879, 569903, 570127, 570212, 570322, 570647, 570702, 570710, 570834, 570865, 570888, 570951, 571301, 571348, 571363, 571377, 571419, 571477, 571479, 571504, 571553, 571638, 571906, 571916, 571959, 572020, 572061, 572082, 572259, 572397, 572504, 572724, 572918, 572923, 573065, 573079, 573250, 573300, 573314, 573317, 573475, 573609, 573780, 573838, 573892, 574033, 574118, 574158, 574340, 574364, 574374, 574422, 574480, 574528, 574670, 574696, 574838, 574887, 574944, 574956, 574994, 575048, 575263, 575441, 575594, 575651, 575812, 575861, 575980, 576054, 576133, 576269, 576283, 576652, 576789, 576854, 576975, 577056, 577252, 577407, 577453, 577539, 57!7672, 577731, 577793, 577809, 577903, 577917, 578118, 578122, 578274, 578590, 579137, 579308, 579380, 579571, 579650, 579661, 579748, 579800, 579868, 580000, 580130, 580431, 580563, 580847, 580865, 580886, 580993, 581138, 581343, 581538, 581685, 581776, 581849, 581904, 582155, 582170, 582345, 582384, 582411, 582548, 582661, 582668, 582716, 582723, 582740, 582988, 583236, 583291, 583360, 583397, 583484, 583532, 583566, 583615, 583688, 583720, 583729, 584097, 584124, 584153, 584192, 584619, 584687, 584778, 584797, 584820, 585028, 585053, 585126, 585323, 585371, 585411, 585434, 585438, 585675, 585931, 585937, 585957, 585962, 586001, 586133, 586166, 586295, 586300, 586436, 586591, 586601, 586910, 586998, 587142, 587192, 587240, 587405, 587419, 587514, 587618, 587707, 587758, 587824, 587865, 587932, 588160, 588214, 588218, 588323, 588378, 588395, 588663, 588677, 588679, 588752, 588808, 588811, 588867, 588887, 589076, 589092, 589187, 589493, 589920, 589946, 590137, 5!90245, 590270, 590283, 591149, 591174, 591188, 591225, 591440, 591459, 591501, 591741, 591827, 591853, 591865, 591886, 592092, 592167, 592188, 592422, 592496, 592565, 592581, 592785, 592808, 593092, 593335, 593488, 593612, 593770, 593862, 593889, 594097, 594103, 594123, 594134, 594236, 594281, 594423, 594456, 594528, 594530, 594540, 594559, 594744, 594757, 594855, 594892, 595141, 595156, 595188, 595206, 595251, 595282, 595451, 595452, 595506, 595648, 595792, 595873, 596067, 596122, 596222, 596401, 596432, 597094, 597254, 597281, 597372, 597429, 597571, 597748, 597869, 597934, 597971, 598078, 598235, 598310, 598354, 598396, 598445, 598555, 598576, 598648, 598828, 598876, 598933, 599072, 599424, 599652, 599751, 599902, 600023, 600118, 600168, 600206, 600649, 600687, 600766, 600888, 601095, 601110, 601144, 601189, 601250, 601321, 601397, 601459, 601626, 601951, 602005, 602017, 602025, 602032, 602081, 602326, 602341, 602401, 602425, 602446, 602524, 602565, 602719, !602729, 602822, 602987, 603042, 603076, 603082, 603241, 603410, 603485, 603599, 603765, 603946, 604109, 604324, 604418, 604583, 604607, 604620, 604627, 604645, 604839, 604844, 605127, 605255, 605405, 605614, 605680, 605820, 606250, 606387, 606427, 606488, 606572, 606642, 606776, 607078, 607186, 607263, 607307, 607312, 607410, 607591, 607625, 607639, 607671, 607673, 607758, 607786, 607844, 607941, 608141, 608193, 608198, 608260, 608337, 608428, 608521, 608531, 608595, 608640, 608786, 608944, 609189, 609744, 609832, 610297, 610480, 610540, 610590, 610601, 610707, 610768, 610795, 611184, 611198, 611228, 611279, 611285, 611458, 611486, 611551, 611553, 611557, 611887, 612078, 612105, 612330, 612522, 612530, 612618, 612627, 612702, 612711, 612723, 612810, 612837, 612850, 612866, 612879, 612892, 612978, 613055, 613070, 613230, 613290, 613411, 613520, 613573, 613630, 613766, 613821, 613867, 613880, 614112, 614114, 614129, 614293, 614303, 614492, 614604, 614717, 614785, !614849, 614858, 615006, 615313, 615375, 615413, 615466, 615477, 615623, 615630, 615647, 615970, 616200, 616390, 616556, 616714, 616717, 616758, 616791, 616796, 616815, 616820, 616833, 617132, 617156, 617208, 617215, 617283, 617304, 617564, 617604, 617803, 617860, 617949, 617958, 618041, 618228, 618273, 618580, 618751, 618835, 618908, 618943, 618952, 619092, 619112, 619198, 619275, 619307, 619477, 619495, 619599, 619610, 619683, 619761, 619807, 619888, 619920, 619952, 619990, 620267, 620298, 620502, 620563, 620684, 620761, 621058, 621206, 621323, 621403, 621558, 621676, 621710, 621851, 622006, 622185, 622190, 622389, 622424, 622450, 622498, 622544, 622550, 622595, 622677, 622859, 622924, 623099, 623200, 623222, 623307, 623315, 623406, 623415, 623443, 623678, 623729, 623762, 623770, 624157, 624272, 624300, 624446, 624506, 624664, 624711, 624757, 624885, 624908, 624937, 625080, 625156, 625181, 625201, 625232, 625235, 625273, 625285, 625478, 625496, 625502, 625543,! 625735, 625746, 625988, 626087, 626172, 626182, 626256, 626511, 626747, 626800, 626826, 626851, 627408, 627497, 627629, 627726, 627863, 628003, 628013, 628207, 628443, 628514, 628548, 628960, 629046, 629048, 629081, 629092, 629127, 629429, 629470, 629553, 629576, 629622, 629778, 629957, 629998, 630131, 630222, 630282, 630297, 630299, 630338, 630393, 630417, 630546, 630707, 630712, 630793, 630818, 631007, 631065, 631067, 631452, 631577, 631580, 631596, 631713, 631907, 631968, 632020, 632128, 632303, 632359, 632429, 632613, 632717, 632746, 632962, 633236, 633246, 633314, 633342, 633541, 633703, 633727, 633783, 633860, 633867, 633872, 634008, 634019, 634028, 634261, 634276, 634320, 634333, 634485, 634614, 634852, 634888, 634895, 634972, 634999, 635016, 635035, 635093, 635152, 635287, 635393, 635539, 635561, 635740, 635796, 635987, 636049, 636064, 636067, 636113, 636133, 636777, 636794, 636856, 636864, 636981, 637057, 637365, 637565, 637623, 637682, 637897, 637966,! 638024, 638110, 638527, 638611, 638650, 638807, 638815, 638845, 638942, 639232, 639324, 639436, 639441, 639444, 639486, 639571, 639722, 639737, 639783, 639894, 639962, 639977, 640057, 640081, 640156, 640190, 640251, 640315, 640389, 640489, 640543, 640555, 640724, 640820, 640834, 641166, 641287, 641300, 641341, 641410, 641425, 641493, 641632, 641741, 641827, 641886, 641915, 641927, 642228, 642270, 642354, 642384, 642534, 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953741, 953753, 953858, 9!53895, 953931, 954343, 954351, 954352, 954401, 954547, 954686, 954706, 955101, 955141, 955194, 955235, 955250, 955353, 955386, 955499, 955664, 955788, 956241, 956330, 956338, 956347, 956580, 956618, 956626, 956642, 956723, 956779, 956837, 956893, 956957, 957105, 957286, 957623, 957682, 957689, 957768, 957792, 957810, 957826, 958112, 958152, 958299, 958357, 958367, 958592, 958747, 958853, 958929, 958993, 959037, 959150, 959273, 959299, 959413, 959438, 959613, 959642, 959646, 959723, 959781, 959814, 959848, 959857, 959900, 960017, 960040, 960103, 960104, 960120, 960151, 960305, 960604, 960666, 960710, 960713, 960729, 960819, 960874, 961033, 961043, 961049, 961125, 961267, 961536, 961670, 961825, 961952, 962091, 962108, 962177, 962183, 962299, 962382, 962432, 962457, 962565, 962909, 962928, 963064, 963239, 963285, 963344, 963411, 963591, 963835, 963871, 964458, 964483, 964730, 964769, 964820, 964825, 964893, 964957, 964979, 965193, 965266, 965277, 965423, 965424, !965633, 965679, 965768, 965905, 965936, 966141, 966258, 966266, 966267, 966383, 966399, 966559, 966579, 966785, 966844, 967044, 967110, 967295, 967307, 967686, 967963, 968024, 968097, 968110, 968113, 968189, 968204, 968208, 968323, 968553, 968617, 968695, 968770, 968808, 968852, 969044, 969207, 969356, 969387, 969442, 969487, 970098, 970127, 970228, 970248, 970309, 970320, 970436, 970458, 970609, 970658, 970722, 970734, 970839, 970911, 970932, 970961, 970981, 970995, 971007, 971028, 971158, 971179, 971383, 971552, 971710, 972033, 972445, 972503, 972526, 972699, 972793, 972971, 973092, 973404, 973466, 973521, 973548, 973584, 973650, 973872, 973998, 974128, 974150, 974349, 974897, 974964, 974997, 975041, 975442, 975443, 975583, 975644, 975660, 975850, 975949, 976122, 976320, 976347, 976349, 976413, 976508, 976603, 976768, 976927, 977006, 977015, 977165, 977192, 977266, 977659, 977781, 977944, 977988, 978059, 978108, 978111, 978279, 978339, 978441, 978572, 978579, ! 978666, 978833, 978838, 978853, 978913, 978936, 979122, 979151, 979192, 979364, 979432, 979521, 979556, 979560, 979591, 979949, 979956, 979979, 980017, 980173, 980186, 980219, 980567, 980586, 980636, 980647, 980728, 980839, 980855, 980872, 981064, 981216, 981236, 981342, 981420, 981528, 981655, 981771, 981847, 981970, 982072, 982389, 982465, 982525, 982543, 982729, 982743, 982756, 982870, 983083, 983136, 983168, 983382, 983560, 983657, 983837, 983891, 984373, 984457, 984544, 984593, 984666, 984673, 984732, 984769, 985001, 985043, 985061, 985226, 985245, 985836, 986005, 986056, 986168, 986234, 986271, 986378, 986466, 986474, 986607, 986629, 986641, 986738, 987026, 987053, 987106, 987170, 987212, 987229, 987293, 987443, 987481, 987652, 987953, 987988, 988006, 988032, 988181, 988188, 988229, 988254, 988276, 988304, 988383, 988553, 988594, 988627, 988668, 989019, 989119, 989130, 989132, 989133, 989462, 989467, 989524, 989654, 989716, 989784, 989800, 989885, 989909,! 990027, 990145, 990194, 990231, 990400, 990429, 990464, 990624, 990824, 990945, 991076, 991087, 991647, 991695, 991775, 991895, 991901, 992087, 992244, 992415, 992419, 992507, 992586, 992751, 992763, 992983, 993005, 993139, 993190, 993194, 993204, 993223, 993381, 993600, 993707, 993727, 993815, 993875, 993914, 994009, 994054, 994157, 994191, 994325, 994349, 994551, 994899, 994924, 995293, 995340, 995450, 995499, 995511, 995525, 995567, 995702, 995879, 995933, 995954, 996127, 996219, 996233, 996368, 996451, 996686, 996733, 996830, 997071, 997105, 997248, 997404, 997434, 997623, 997642, 997695, 997726, 997754, 997869, 998008, 998123, 998190, 998196, 998215, 998431, 998493, 998656, 998688, 998695, 998789, 998893, 998942, 999058, 999117, 999556, 999795, 999859, 999899, 999966, 999979}=== === Subject: : Re: Fibonacci PHI, PI, e Relation by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i0P1Sor07515;The forum did not let me post it all. Here is some more419997, 420018, 420085, 420232, 420235, 420470, 420507, 420589, 420691, 421144, 421339, 421456, 421485, 421498, 421699, 421854, 421974, 421988, 422042, 422045, 422194, 422412, 422472, 422551, 422828, 422917, 422995, 423053, 423405, 423503, 423535, 423706, 423760, 423892, 424057, 424089, 424128, 424154, 424220, 424318, 424395, 424459, 424494, 424513, 424529, 424675, 424711, 425067, 425131, 425148, 425304, 425363, 425620, 425820, 425920, 426292, 426464, 426490, 426594, 426646, 426653, 426735, 426889, 426930, 426936, 427088, 427133, 427230, 427352, 427517, 427849, 427930, 427972, 428021, 428026, 428039, 428168, 428189, 428219, 428230, 428250, 428270, 428322, 428324, 428328, 428711, 428748, 428910, 429030, 429058, 429167, 429170, 429183, 429434, 429841, 429915, 429925, 430168, 430206, 430232, 430346, 430524, 430605, 430625, 430818, 431098, 431185, 431230, 431254, 431408, 431649, 431708, 431767, 431846, 431931, 431961, 432309, 432377, 432421, 432544, 432561, 432688, !432741, 432872, 432926, 432937, 432997, 433034, 433297, 433434, 433718, 433800, 433903, 433913, 434062, 434081, 434082, 434260, 434614, 434632, 434754, 434819, 434875, 434891, 434938, 435095, 435103, 435139, 435194, 435244, 435292, 435378, 435443, 435604, 435649, 435750, 436166, 436207, 436278, 436367, 436368, 436372, 436543, 436562, 436594, 436678, 436682, 436893, 436983, 437009, 437029, 437114, 437206, 437229, 437281, 437387, 437569, 437606, 437665, 437744, 437758, 437980, 438085, 438122, 438232, 438442, 438677, 438707, 438744, 438890, 438905, 438979, 439066, 439119, 439233, 439258, 439319, 439498, 439691, 439797, 440403, 441099, 441175, 441264, 441272, 441312, 441353, 441407, 441561, 441767, 442048, 442111, 442141, 442160, 442353, 442374, 442390, 442407, 442416, 442418, 442423, 442572, 443084, 443097, 443224, 443435, 443444, 443554, 443557, 443656, 443855, 443875, 443896, 443945, 443996, 444310, 444353, 444469, 444626, 445040, 445164, 445179, 445187, 445303,! 445329, 445352, 445501, 445540, 445541, 445554, 445600, 445620, 445797, 445839, 445976, 446202, 446359, 446517, 446622, 446700, 446709, 446872, 446975, 446988, 447025, 447082, 447097, 447361, 447478, 447579, 447744, 447780, 448134, 448182, 448215, 448277, 448302, 448367, 448584, 448614, 448712, 448858, 449017, 449090, 449198, 449381, 449600, 449738, 449750, 449880, 449909, 449921, 450038, 450083, 450142, 450183, 450198, 450353, 450666, 450813, 450892, 450893, 451070, 451157, 451182, 451431, 451592, 451674, 451797, 451859, 451958, 452208, 452209, 452335, 452390, 452410, 452526, 452563, 452681, 452708, 452739, 452802, 452894, 453025, 453073, 453219, 453372, 453454, 453482, 453488, 453528, 453592, 453849, 453880, 453974, 454181, 454343, 454379, 454384, 454423, 454552, 454626, 454661, 454751, 454784, 454908, 455022, 455049, 455123, 455138, 455254, 455267, 455441, 455443, 455468, 455604, 455613, 455663, 456041, 456262, 456357, 456369, 456375, 456430, 456459, 456574,! 456709, 456791, 456808, 456843, 456857, 456983, 457068, 457123, 457321, 457618, 457767, 457829, 457931, 458057, 458296, 458553, 458755, 458863, 458993, 458998, 459343, 459397, 459450, 459647, 459750, 459754, 459825, 459849, 459894, 459942, 459947, 460093, 460104, 460292, 460334, 460412, 460642, 460686, 461020, 461151, 461164, 461297, 461350, 461390, 461495, 461540, 461693, 461738, 461801, 461839, 461864, 461866, 461941, 461975, 462072, 462132, 462133, 462155, 462350, 462375, 462462, 462548, 462898, 462921, 462988, 463165, 463339, 463413, 463416, 463446, 463473, 463532, 463574, 463769, 463959, 464062, 464088, 464093, 464245, 464271, 464277, 464291, 464367, 464429, 464517, 464797, 464851, 464873, 464880, 464948, 465153, 465155, 465199, 465247, 465282, 465448, 465768, 465823, 465957, 466064, 466124, 466341, 466410, 466412, 466599, 466610, 466617, 466651, 466663, 466734, 466738, 466884, 466887, 467019, 467146, 467213, 467450, 467599, 467646, 467741, 467864, 467900!, 468028, 468074, 468084, 468138, 468153, 468458, 468501, 468559, 468565, 468688, 468740, 469001, 469489, 469699, 469752, 469959, 469966, 469989, 470150, 470218, 470314, 470347, 470379, 470383, 470624, 470668, 470753, 470757, 470788, 470814, 471100, 471165, 471172, 471270, 471436, 471437, 471467, 471470, 471525, 471537, 471659, 471732, 471820, 471935, 472050, 472064, 472217, 472315, 472420, 472546, 472689, 472691, 472790, 472856, 473135, 473185, 473906, 473949, 473998, 474007, 474023, 474080, 474131, 474161, 474209, 474475, 474557, 474749, 474854, 474873, 474896, 475038, 475092, 475095, 475275, 475317, 475462, 475540, 475668, 475683, 475865, 475909, 476028, 476091, 476112, 476149, 476217, 476245, 476279, 476296, 476386, 476510, 476677, 476812, 476825, 476854, 477192, 477247, 477277, 477291, 477293, 477320, 477498, 477522, 477602, 477620, 477844, 477907, 478045, 478214, 478226, 478246, 478374, 478439, 478520, 478524, 478592, 478764, 478798, 478835, 479058, 479116!, 479242, 479269, 479279, 479387, 479483, 479693, 479700, 479771, 479855, 479967, 480054, 480187, 480612, 480629, 480735, 480860, 481285, 481410, 481470, 481471, 481526, 481640, 481769, 481795, 481838, 481856, 481937, 482080, 482104, 482118, 482662, 482667, 482682, 482827, 483000, 483019, 483124, 483131, 483269, 483378, 483388, 483431, 483452, 483486, 483622, 483765, 483894, 484063, 484241, 484310, 484392, 484399, 484755, 484779, 484889, 485132, 485206, 485317, 485434, 485481, 485568, 485753, 485964, 486061, 486136, 486351, 486390, 486493, 486539, 486557, 486594, 486609, 486741, 486743, 486990, 487171, 487176, 487225, 487493, 487520, 487556, 487592, 487821, 487868, 487914, 488078, 488329, 488379, 488440, 488468, 489193, 489399, 489536, 489703, 489763, 489834, 489901, 489929, 490037, 490172, 490225, 490335, 490398, 490404, 490433, 490520, 490563, 490601, 490713, 490728, 490762, 490791, 490803, 490841, 490900, 490941, 491061, 491179, 491223, 491397, 491654, 49171!2, 491849, 492158, 492307, 492570, 492594, 492877, 492983, 493040, 493077, 493320, 493340, 493480, 493533, 493570, 493571, 493652, 493661, 493740, 493761, 493762, 493814, 494044, 494149, 494337, 494395, 494477, 494627, 494652, 494671, 494755, 494768, 494835, 494906, 494958, 495000, 495203, 495324, 495332, 495400, 495410, 495586, 495597, 495751, 495897, 495947, 496038, 496187, 496207, 496312, 496411, 496421, 496497, 496566, 496859, 496883, 496904, 497023, 497099, 497265, 497387, 497406, 497464, 497483, 497539, 497687, 497731, 498005, 498046, 498116, 498318, 498326, 498442, 498470, 498644, 498670, 498841, 499060, 499169, 499320, 499459, 499554, 499594, 499749, 499851, 500090, 500278, 500394, 500439, 500459, 500503, 500749, 500771, 500792, 500890, 500944, 500970, 501000, 501297, 501333, 501460, 501538, 501654, 501657, 501692, 501768, 501812, 501852, 501930, 501995, 502230, 502607, 503142, 503259, 503268, 503272, 503315, 503360, 503430, 503468, 503525, 503537, 50358!4, 503586, 503663, 503737, 503809, 503839, 503847, 503899, 504032, 504079, 504264, 504477, 504490, 504639, 504696, 504869, 504880, 504909, 504918, 504922, 505148, 505223, 505250, 505452, 505533, 505589, 505641, 505727, 505733, 505766, 505979, 506181, 506344, 506359, 506399, 506599, 506655, 506670, 506748, 506815, 506821, 506965, 506973, 506980, 507052, 507072, 507183, 507231, 507249, 507424, 507530, 507876, 507889, 507911, 508039, 508080, 508395, 508554, 508628, 508763, 508978, 508993, 509022, 509288, 509297, 509370, 509375, 509491, 509554, 509615, 509729, 509803, 510106, 510107, 510229, 510416, 510456, 510748, 510796, 510912, 511015, 511107, 511120, 511129, 511266, 511532, 511554, 511956, 512160, 512213, 512343, 512350, 512478, 512554, 512585, 512624, 512729, 512966, 513032, 513067, 513089, 513133, 513396, 513402, 513427, 513515, 513624, 513626, 513796, 513812, 513996, 514038, 514099, 514388, 514466, 514576, 514644, 514714, 514782, 514805, 514948, 515010, 5151!39, 515141, 515146, 515305, 515522, 515576, 515646, 515659, 515802, 515981, 516010, 516025, 516120, 516126, 516190, 516302, 516393, 516542, 516581, 516694, 516834, 516846, 517085, 517126, 517222, 517298, 517528, 517634, 517648, 517678, 517863, 518151, 518155, 518181, 518245, 518256, 518279, 518286, 518403, 518499, 518714, 518728, 518753, 518822, 518926, 518943, 518962, 519099, 519284, 519298, 519310, 519391, 519503, 519572, 519582, 519585, 519634, 519670, 519823, 519904, 520128, 520135, 520171, 520229, 520294, 520562, 520656, 520788, 520794, 520874, 520883, 521106, 521249, 521281, 521293, 521383, 521491, 521511, 521648, 521696, 521798, 521868, 521950, 522107, 522217, 522258, 522554, 522627, 522890, 523042, 523102, 523434, 523451, 523621, 523723, 523791, 523795, 523900, 523911, 523931, 524013, 524073, 524147, 524182, 524304, 524584, 524805, 524845, 525028, 525092, 525327, 525358, 525388, 525443, 525480, 525612, 525620, 525765, 525768, 525932, 526012, 526134, 5262!09, 526310, 526419, 526671, 526752, 526854, 526878, 526883, 526891, 526906, 526959, 527073, 527330, 527447, 527533, 527564, 527579, 527605, 527635, 527903, 528071, 528211, 528320, 528508, 528587, 528602, 528645, 528702, 528878, 528917, 529000, 529087, 529106, 529195, 529436, 529508, 529617, 529651, 529682, 529923, 530047, 530097, 530130, 530510, 530550, 530627, 530673, 530843, 530884, 530892, 531182, 531350, 531365, 531373, 531528, 531546, 531580, 531639, 531641, 531768, 531921, 531933, 531979, 532001, 532031, 532095, 532240, 532282, 532400, 532574, 532732, 532836, 532949, 532951, 532996, 533103, 533145, 533251, 533261, 533374, 533395, 533431, 533466, 533513, 533687, 533882, 533961, 533995, 534070, 534132, 534189, 534192, 534287, 534291, 534399, 534467, 534667, 534788, 534850, 534852, 535093, 535105, 535112, 535277, 535311, 535459, 535467, 535745, 535762, 535849, 535877, 536093, 536110, 536231, 536270, 536421, 536534, 536626, 536867, 536969, 536989, 537029, 537!047, 537104, 537143, 537168, 537242, 537382, 537426, 537481, 537654, 537696, 537763, 537769, 537961, 537990, 538007, 538027, 538035, 538172, 538210, 538282, 538365, 538370, 538401, 538716, 538777, 538905, 539059, 539183, 539369, 539425, 539523, 539532, 539548, 539631, 539712, 539830, 539928, 539976, 539997, 540012, 540174, 540430, 541164, 541451, 541567, 541589, 541670, 541886, 542011, 542235, 542293, 542516, 542621, 542740, 543430, 543461, 543580, 543708, 543992, 544008, 544016, 544179, 544253, 544279, 544423, 544466, 544488, 544528, 544599, 544672, 544785, 544980, 545024, 545527, 545792, 545810, 545831, 545835, 545887, 545935, 546004, 546006, 546035, 546046, 546163, 546217, 546237, 546238, 546273, 546497, 546633, 546716, 546796, 546805, 546886, 547149, 547244, 547363, 547394, 547608, 547802, 547851, 548139, 548140, 548172, 548245, 548417, 548481, 548584, 548729, 548940, 549154, 549361, 549491, 549594, 549675, 549754, 549837, 550104, 550206, 550304, 550384, 55!0468, 550529, 550644, 550703, 550746, 551039, 551057, 551096, 551279, 551321, 551324, 551350, 551395, 551453, 551867, 552043, 552120, 552180, 552233, 552273, 552463, 552482, 552542, 552672, 552861, 553090, 553155, 553205, 553307, 553337, 553387, 553523, 553593, 553644, 553721, 553802, 553843, 554031, 554151, 554270, 554626, 554724, 554769, 554813, 554871, 554950, 555198, 555258, 555305, 555381, 555538, 556299, 556311, 556419, 556510, 556599, 556729, 556793, 556913, 556921, 556940, 557017, 557073, 557288, 557569, 557653, 558174, 558219, 558378, 558611, 558675, 558782, 558933, 559017, 559021, 559107, 559170, 559184, 559421, 559486, 559596, 559680, 559761, 559763, 559863, 559967, 560506, 560732, 560980, 561290, 561398, 561409, 561619, 561895, 561985, 562058, 562071, 562243, 562263, 562282, 562289, 562611, 562649, 562676, 562856, 562895, 563095, 563384, 563413, 563446, 563538, 563613, 563621, 563626, 563683, 563728, 563744, 563772, 563794, 563848, 564097, 564099, 56!4217, 564772, 564970, 564989, 565030, 565092, 565230, 566012, 566188, 566251, 566301, 566316, 566600, 566615, 566698, 566723, 566895, 567077, 567092, 567098, 567131, 567143, 567210, 567257, 567658, 567706, 567878, 567939, 568219, 568252, 568298, 568370, 568381, 568426, 568553, 568665, 569136, 569370, 569396, 569407, 569435, 569511, 569747, 569879, 569903, 570127, 570212, 570322, 570647, 570702, 570710, 570834, 570865, 570888, 570951, 571301, 571348, 571363, 571377, 571419, 571477, 571479, 571504, 571553, 571638, 571906, 571916, 571959, 572020, 572061, 572082, 572259, 572397, 572504, 572724, 572918, 572923, 573065, 573079, 573250, 573300, 573314, 573317, 573475, 573609, 573780, 573838, 573892, 574033, 574118, 574158, 574340, 574364, 574374, 574422, 574480, 574528, 574670, 574696, 574838, 574887, 574944, 574956, 574994, 575048, 575263, 575441, 575594, 575651, 575812, 575861, 575980, 576054, 576133, 576269, 576283, 576652, 576789, 576854, 576975, 577056, 577252, 5!77407, 577453, 577539, 577672, 577731, 577793, 577809, 577903, 577917, 578118, 578122, 578274, 578590, 579137, 579308, 579380, 579571, 579650, 579661, 579748, 579800, 579868, 580000, 580130, 580431, 580563, 580847, 580865, 580886, 580993, 581138, 581343, 581538, 581685, 581776, 581849, 581904, 582155, 582170, 582345, 582384, 582411, 582548, 582661, 582668, 582716, 582723, 582740, 582988, 583236, 583291, 583360, 583397, 583484, 583532, 583566, 583615, 583688, 583720, 583729, 584097, 584124, 584153, 584192, 584619, 584687, 584778, 584797, 584820, 585028, 585053, 585126, 585323, 585371, 585411, 585434, 585438, 585675, 585931, 585937, 585957, 585962, 586001, 586133, 586166, 586295, 586300, 586436, 586591, 586601, 586910, 586998, 587142, 587192, 587240, 587405, 587419, 587514, 587618, 587707, 587758, 587824, 587865, 587932, 588160, 588214, 588218, 588323, 588378, 588395, 588663, 588677, 588679, 588752, 588808, 588811, 588867, 588887, 589076, 589092, 589187, 589493, 5!89920, 589946, 590137, 590245, 590270, 590283, 591149, 591174, 591188, 591225, 591440, 591459, 591501, 591741, 591827, 591853, 591865, 591886, 592092, 592167, 592188, 592422, 592496, 592565, 592581, 592785, 592808, 593092, 593335, 593488, 593612, 593770, 593862, 593889, 594097, 594103, 594123, 594134, 594236, 594281, 594423, 594456, 594528, 594530, 594540, 594559, 594744, 594757, 594855, 594892, 595141, 595156, 595188, 595206, 595251, 595282, 595451, 595452, 595506, 595648, 595792, 595873, 596067, 596122, 596222, 596401, 596432, 597094, 597254, 597281, 597372, 597429, 597571, 597748, 597869, 597934, 597971, 598078, 598235, 598310, 598354, 598396, 598445, 598555, 598576, 598648, 598828, 598876, 598933, 599072, 599424, 599652, 599751, 599902, 600023, 600118, 600168, 600206, 600649, 600687, 600766, 600888, 601095, 601110, 601144, 601189, 601250, 601321, 601397, 601459, 601626, 601951, 602005, 602017, 602025, 602032, 602081, 602326, 602341, 602401, 602425, 602446, !602524, 602565, 602719, 602729, 602822, 602987, 603042, 603076, 603082, 603241, 603410, 603485, 603599, 603765, 603946, 604109, 604324, 604418, 604583, 604607, 604620, 604627, 604645, 604839, 604844, 605127, 605255, 605405, 605614, 605680, 605820, 606250, 606387, 606427, 606488, 606572, 606642, 606776, 607078, 607186, 607263, 607307, 607312, 607410, 607591, 607625, 607639, 607671, 607673, 607758, 607786, 607844, 607941, 608141, 608193, 608198, 608260, 608337, 608428, 608521, 608531, 608595, 608640, 608786, 608944, 609189, 609744, 609832, 610297, 610480, 610540, 610590, 610601, 610707, 610768, 610795, 611184, 611198, 611228, 611279, 611285, 611458, 611486, 611551, 611553, 611557, 611887, 612078, 612105, 612330, 612522, 612530, 612618, 612627, 612702, 612711, 612723, 612810, 612837, 612850, 612866, 612879, 612892, 612978, 613055, 613070, 613230, 613290, 613411, 613520, 613573, 613630, 613766, 613821, 613867, 613880, 614112, 614114, 614129, 614293, 614303, 614492, !614604, 614717, 614785, 614849, 614858, 615006, 615313, 615375, 615413, 615466, 615477, 615623, 615630, 615647, 615970, 616200, 616390, 616556, 616714, 616717, 616758, 616791, 616796, 616815, 616820, 616833, 617132, 617156, 617208, 617215, 617283, 617304, 617564, 617604, 617803, 617860, 617949, 617958, 618041, 618228, 618273, 618580, 618751, 618835, 618908, 618943, 618952, 619092, 619112, 619198, 619275, 619307, 619477, 619495, 619599, 619610, 619683, 619761, 619807, 619888, 619920, 619952, 619990, 620267, 620298, 620502, 620563, 620684, 620761, 621058, 621206, 621323, 621403, 621558, 621676, 621710, 621851, 622006, 622185, 622190, 622389, 622424, 622450, 622498, 622544, 622550, 622595, 622677, 622859, 622924, 623099, 623200, 623222, 623307, 623315, 623406, 623415, 623443, 623678, 623729, 623762, 623770, 624157, 624272, 624300, 624446, 624506, 624664, 624711, 624757, 624885, 624908, 624937, 625080, 625156, 625181, 625201, 625232, 625235, 625273, 625285, 625478,! 625496, 625502, 625543, 625735, 625746, 625988, 626087, 626172, 626182, 626256, 626511, 626747, 626800, 626826, 626851, 627408, 627497, 627629, 627726, 627863, 628003, 628013, 628207, 628443, 628514, 628548, 628960, 629046, 629048, 629081, 629092, 629127, 629429, 629470, 629553, 629576, 629622, 629778, 629957, 629998, 630131, 630222, 630282, 630297, 630299, 630338, 630393, 630417, 630546, 630707, 630712, 630793, 630818, 631007, 631065, 631067, 631452, 631577, 631580, 631596, 631713, 631907, 631968, 632020, 632128, 632303, 632359, 632429, 632613, 632717, 632746, 632962, 633236, 633246, 633314, 633342, 633541, 633703, 633727, 633783, 633860, 633867, 633872, 634008, 634019, 634028, 634261, 634276, 634320, 634333, 634485, 634614, 634852, 634888, 634895, 634972, 634999, 635016, 635035, 635093, 635152, 635287, 635393, 635539, 635561, 635740, 635796, 635987, 636049, 636064, 636067, 636113, 636133, 636777, 636794, 636856, 636864, 636981, 637057, 637365, 637565, 637623,! 637682, 637897, 637966, 638024, 638110, 638527, 638611, 638650, 638807, 638815, 638845, 638942, 639232, 639324, 639436, 639441, 639444, 639486, 639571, 639722, 639737, 639783, 639894, 639962, 639977, 640057, 640081, 640156, 640190, 640251, 640315, 640389, 640489, 640543, 640555, 640724, 640820, 640834, 641166, 641287, 641300, 641341, 641410, 641425, 641493, 641632, 641741, 641827, 641886, 641915, 641927, 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978339,! 978441, 978572, 978579, 978666, 978833, 978838, 978853, 978913, 978936, 979122, 979151, 979192, 979364, 979432, 979521, 979556, 979560, 979591, 979949, 979956, 979979, 980017, 980173, 980186, 980219, 980567, 980586, 980636, 980647, 980728, 980839, 980855, 980872, 981064, 981216, 981236, 981342, 981420, 981528, 981655, 981771, 981847, 981970, 982072, 982389, 982465, 982525, 982543, 982729, 982743, 982756, 982870, 983083, 983136, 983168, 983382, 983560, 983657, 983837, 983891, 984373, 984457, 984544, 984593, 984666, 984673, 984732, 984769, 985001, 985043, 985061, 985226, 985245, 985836, 986005, 986056, 986168, 986234, 986271, 986378, 986466, 986474, 986607, 986629, 986641, 986738, 987026, 987053, 987106, 987170, 987212, 987229, 987293, 987443, 987481, 987652, 987953, 987988, 988006, 988032, 988181, 988188, 988229, 988254, 988276, 988304, 988383, 988553, 988594, 988627, 988668, 989019, 989119, 989130, 989132, 989133, 989462, 989467, 989524, 989654, 989716, 989784,! 989800, 989885, 989909, 990027, 990145, 990194, 990231, 990400, 990429, 990464, 990624, 990824, 990945, 991076, 991087, 991647, 991695, 991775, 991895, 991901, 992087, 992244, 992415, 992419, 992507, 992586, 992751, 992763, 992983, 993005, 993139, 993190, 993194, 993204, 993223, 993381, 993600, 993707, 993727, 993815, 993875, 993914, 994009, 994054, 994157, 994191, 994325, 994349, 994551, 994899, 994924, 995293, 995340, 995450, 995499, 995511, 995525, 995567, 995702, 995879, 995933, 995954, 996127, 996219, 996233, 996368, 996451, 996686, 996733, 996830, 997071, 997105, 997248, 997404, 997434, 997623, 997642, 997695, 997726, 997754, 997869, 998008, 998123, 998190, 998196, 998215, 998431, 998493, 998656, 998688, 998695, 998789, 998893, 998942, 999058, 999117, 999556, 999795, 999859, 999899, 999966, 999979}=== === Subject: : Re: Factoring Conjecture: Triplets>Subject: Re: Factoring Conjecture: Triplets>Message-id: I agree that he's disturbed, but not the way you think.>I'm intrigued. What kind of disturbed do you think he is?A danger to himself and others.>Gib--=== === Subject: : Re: general polyhedronI *did* forget about thickness;the problem of shingling can be mapped to good, ole 2D. as for your novel conjectureabout the difference between a dodecasteron (icosahedron) anda dodecaXIasteron (cuboctahedron),How in Hell is that? solid angle is only the same as a measure of areafor the sphere. > Right, that's the question. Now, solid angle is a measure of area, so> before going further we need to say which configuration we are going to> look at for a poly with some number of vertexes. In other words, no> poly with more that 4 vertexes will have the same total solid angle> for the different possible arrangements of the vertexes. Yes?> For example, an icosahedron and a cubeoctahedron both have 12 vertexes> and each of the 12 vertexes have an angular deficit of 60 degrees. But> the two configutations do not have the same area(solid angle).--ils duces d'Enron!http://tarpley.net=== === Subject: : Re: Rationals are Uncountable> Let X be the set of rational numbers in (0,1).> Let x_n be a sequence of rational numbers> that contains every member of X.> I have previously described such a sequence.> Construct sequences a_n and b_n using> Cantor's algorithm.> Are the sequences a_n and b_n finite?NOT if done as Cantor described.> Assume they are finite.Why? > There exists an interval (a_i, b_i)> such that x_n contains no rationals> in this interval.Since we have star with a necessarily dense set, if we follow Cantor, there cannot be any two members of that set with no members of the set between them. This holds for the rationals.> Assume a_n and b_n are infinite.> There exists an infinite number of> rational numbers in X not in x_n.This is not necessarily true, as surjections from N to Q are known. if the sequence x_n is one of those surjections onto Q, then there are NO rationals not in the sequence x_n.> Russell- 2 stupid 2 count=== === Subject: : Re: Rationals are Uncountable> Let X be the set of rational numbers in (0,1).> Let x_n be a sequence of rational numbers> that contains every member of X.> I have previously described such a sequence.Construct sequences a_n and b_n using> Cantor's algorithm.Are the sequences a_n and b_n finite?> NOT if done as Cantor described.Assume they are finite.> Why?To eliminate the possibility.> There exists an interval (a_i, b_i)> such that x_n contains no rationals> in this interval.> Since we have star with a necessarily dense set, if we follow Cantor,> there cannot be any two members of that set with no members of the set> between them. This holds for the rationals.Agreed.Sequences A and B can never be finite if the rationals are dense.> Assume a_n and b_n are infinite.> There exists an infinite number of> rational numbers in X not in x_n.> This is not necessarily true, as surjections from N to Q are known. if> the sequence x_n is one of those surjections onto Q, then there are NO> rationals not in the sequence x_n.>The point of the proof is to show no such surjections exists.The wikipedia proof states:The claim is that c cannot be in the range of the sequence x, and that isthe contradiction. If c were in the range, then we would have c = xi forsome index i. But then, when that index was reached in the process ofdefining a and b, then c would have been added as the next member of one orthe other of those two sequences, contrary to the assumption that it liesbetween their ranges.Nothing changes if we assume c is rational.The assumption that the rationalsare dense is sufficient to prove c must exist.Let a_n = x_i and b_n = x_j.Rewrite the interval as (x_i, x_j)For every x_j there exists an x_i, i There are no values between zero and iota because that's iota's> definition, a minimally positive value, next in the increasing> sequence of ordered real numbers after zero.> I could define iota as the leader of the little blue people who live> in Ross's walls. That would not entitle me to proceed as if iota> actually exis.> A Google(tm) search for iota infinitesimal provides some references> to the consideration of iota as an infinitesimal quantity,> particularly with its quality of being the least real quantity.The least real quantity, if it were to exist,would not be close to zero, though its reciprocal might be.And if iota should exist as the least real positive quantity, how big is iota/2?=== === Subject: : Re: Action Device: Destination Alpha Centauri A> Look, until I plan my future life, I can not build this Action Device.> Alpha Centauri A is 4.35 ly away. So, please tell me, I am> accelerating my bike in space at the rate of 1g i.e. 9.8 m/s^2. How> much time it will take to cover 2.18 lightyears distance i.e. when I> have covered half the road?> Please Note: Velocity Does Not Matter....> Thanks!> -Abhi.Plan for about 1.79 years. Bon voyage!=== === Subject: : Re: Action Device: Destination Alpha Centauri A> Look, until I plan my future life, I can not build this Action Device.> I am precognitive -- I predic you would continue to post about it!> -=-=-=-=-I am also precognitive -- I predic you would reply to me!-Abhi.=== === Subject: : Re: Action Device: Destination Alpha Centauri A> Look, until I plan my future life, I can not build this Action Device.> Alpha Centauri A is 4.35 ly away. So, please tell me, I am> accelerating my bike in space at the rate of 1g i.e. 9.8 m/s^2. How> much time it will take to cover 2.18 lightyears distance i.e. when I> have covered half the road?Who holds the watch, idiot wog?http://fourmilab.to/etexts/einstein/specrel/specrel.pdf< http://www.geocities.com/physics_world/sr/ae_1905_error.htm>-- Uncle Alhttp://www.mazepath.com/uncleal/qz.pdfhttp:// www.mazepath.com/uncleal/eotvos.htm (Do something naughty to physics) === Subject: : re:Hilbert space problems===Are these problems hard?----== Pos via Newsfeed.Com - Unlimi-Uncensored-Secure Usenet News==----http://www.newsfeed.com The #1 Newsgroup Service in the World! >100,000 Newsgroups---= 19 East/West-Coast Specialized Servers - Total Privacy via Encryption =---=== === Subject: : Re: Rational LimitsCan you give me an element in this set?If there are infinitely many, it shouldn't be hard to name just one.That's not entirely fair.The proof that there are infinitely many uncomputable functions is> considerably simpler than exhibiting a particular uncomputable> function (not that the proof that halting problem is uncomputable is> particularly hard, but it's harder than proving there are uncountably> many uncomputable functions in my estimation).On the other hand, we *do* have a proof that the intersection of> {(0, 1/n) | n in N} is empty, so he should either (1) find an explicit> flaw in that proof or (2) find a valid proof of the negation of that> claim.Or he can go on saying, Well, I think .... and you expect me to> believe ...? Very compelling, that line is.> I want to give a variation of Cantor's proof:It would help if you understood the original version before trying to play such idiot variational games.> Let X be the set of all real numbers in [0,1].> Assume there exists a sequence, x_n, that> includes every member of X.> Construct two new sequences, a_n and b_n.> a_n = 0> b_n = x_m if x_m is of the form 1/k (k integer) and x_m < b_(n-1)And what is b_n if x_m is not of the above form?> Is b_n a finite sequence?If you are asking whether one can pick out of the x_n sequence in sequential order infinitely many terms of form 1/m each smaller than all the previously chosen ones, the answer is yes.> Assume b_n is finite.Can't happen.> This means there exists an n such that (a_n, b_n) is the empty set.Can 't happen, for a lot of reasons.> We assumed x_n contains every member of X.> a_n and b_n are non-equal rational numbers.> There must be an infinite number of rationals not in x_n.It does not follow.> Assume b_n is infinite.> This is the basis of Cantor's proof.Not so. The a_n in Cantor's proof must all be different but you have them all the same. This may seem like a minor detail to you, but that is because you do not have any idea of what is really going on here.> If b_n is infinite, there are an infinite number of reals> not in the sequence x_n.Since the x_n are all rationals in your construction, there are uncountably many reals not in the sequence x_n.You are not in the same county as Cantor's proof, not even in the same state, probably not even in the same universe.> How does this relate to the intersection of (0,1/n)?> Proofs have been given showing the intersection of> (0,1/n) must be empty. Here is an example.> For any real x there is an n such that x > 1/n.> Therefore, there can be no x such that 1/n > x for all n.Get it right, for any POSITIVE real x.> Compare that to this:> For any natural, n, there is a real x such that 1/n > x.> Therefore, there can be no n such that x > 1/n for all x.Comparison made. The first makes sense as correc, the second doesn't even come close. The fact that more or less the same words were used in both is not enough.You are scrambling your quantifiers.For all x there is a y does not mean at all the same as for all y there is an x. It is commonly the cse that one form will leas to a true statement and teh other to a false statement.> Let X be the set of rational number in [0,1].> Let x_n be a sequence that includes every member of X.> Many examples of x_n have been given in other threads.> Applying Cantor's algorithm, will b_n be a finite sequence?If you do it as Cantor requires it done, neither the a_n nor the b_n consists of only finitely many values.If Cantor says go East on Interstate 80 and you will get to Chicago, you would read him as saying Interstae 70 will gat you to Chicgo, and tTHEN complaining that your reading gives a false result.What Canor says in that theoem is so. Live with it or die from it, but stop pissing on what you cannot understand.> Assume b_n is finite. This proves x_n does not contain any rationals> between (a_n, b_n) for some n.An assumption proves nothing.> Assume b_n is infinite. There must be an infinite number of> rationals not in the range of x_n.Why? If your x_m sequence covers all of Q/[0,1], then there will be an infinite sequence of b_n = 1/m in increasing order of both n and m.> Saying the intersection of (0,1/n) is empty is equivalent to saying> the reals are countable.Not at all, Russell. You goofed again.Saying that that intersection is NOT empty is saying that there must be some fixed real (possibly rational) number, say r, in some standard model of the reals numbers, such that 0 < r < 1/n for EVERY positive integer n.Russell, you can NOT produce such a number, nor produce any evidence that such a number can, or even oough to, exist. So Russell, you are blowing smoke again. If you can't stand the rigor, get out of math.=== === Subject: : Re: functions neededSince you're dealing with integers, you can form real numbers of the formf(a,b) = a.bFor instance, f(17,23) = 17.23> I need to construct such a family of functions which> satisfy following constrains:> f(a,b) = f(c,d) if and only if a=b, c=d or a=c, b=d.> for all positive integers a,b,c,d.> Thanks=== === Subject: : Re: Past posters, consider facts>> What if you were faced with arguments over the existence of sqrt(2),> or sqrt(-1)?> Imagine if as a group mathematicians had risen to protect the status> quo, and rejec sqrt(-1)?sqrt(-1), sqrt(2), and a host of other things we now use today at one timewere not accep by the mathematicians of the day. BUT, in math, the truthalways comes out eventually, so you have nothing to worry about. YOU mightnot be able to convince anybody of anything (as indeed appears to be thecase), BUT if you are right, eventually someoe else will be able to. (ohBTW, history shows that railing against the establishment is not a veryeffective method, so your probably accomplishing even less with your rantsand fits).Comforting thought, eh?Jack=== === Subject: : Re: Past posters, consider facts> BTW, history shows that railing against the establishment is not a very> effective method, so your probably accomplishing even less with yourrants> and fits).No, he's accomplishing quite a bit, even though what he's accomplishing isorthogonal to what he's try to accomplish.=== === Subject: : What kind of problem does this belong to? error analysis for matrix-vector multiplication?Dear all,I am facing with the following problem: I have a square matrix A(NxN), and a column vector x(Nx1), By computing y=A*x, I get another column vecotr y(Nx1). But in implementation, due to some error, some small error wasintroduced and actually I compu:y=(A+E)*x, where E is a small error matrix(NxN), now I want to analyze which element of y is impac most heavily, andin what degree, and also want to have an order/ranking of the elementsof y being affec...I am not sure whether this kind of analysis can be made independent ofx or not... if it should be dependent on x, then what characteristicsof x combined with what kind of E gives most impact to which elementof y?Thank you very much in advance!-Walala=== === Subject: : Re: Socrates' Nothing is Everything> Zero is ubiquitous in that arises from the operation a-a = 0.> Zero has meaning in real experience as absence or emptiness.> In the context of physics it is rather out of place as far as the> conservation of mass/energy is concerned but lives hapily with thoseThat operation is only well defined for numbers. Zero, the number, is> not absence or emptiness.> What you should have said is that zero does not signify absence or> emptiness.> What I meant was that nought, nothing, zero and the like do refer to absence> and emptiness> in ordinary speech and in the practical use of arithmetic. For example, if> there are no packets of cornflakes on a supermaket shelf the number zero> will be entered in the stock sheets..> If your wallet is empty you have zero cash. Such obvuious meanings hardly> warrent comment.> As to the operation a-a, if you spend all the cash you have, this expresses> what happened in numbers. So, if you spent 2 dollars, the expression would> be 2-2 = 0 and if you spent ten dollars it would be 10-10 = 0. These> statements represent different events.> Zero is a number, a token of a different> type than absence or emptiness. And I would suggest you don't> take modern physics too seriously. It is notorious for using> suggestive terminology which has little to do with the physical> meaning.>Metaphysically, zero can be considered fundamental or its negation. In> spatial terms this is infinite extension in infinite dimensions. In> physical> terms it would be an infinitely dense mass of infinite extension ie the> plenum void. In arithmetic it would be absolute infinity, that most> intractable of all entities.This isn't metaphysics. It's barely coherent english.> I was being a bit loose here. The process of negation has a dual role, that> of nullifying and that of transforming. These roles can be clarrified by> considering four monadic operators:> (1) N00: nullification> (2) N01: affirmation or stasis> (3) N10: negation> (4) N11: plenum nullification> N00x -> 00> N01x -> x> N10x -> ~x> N11 -> 11> In its arithmetical context zero acts as type (2) with respect to addition,> type (1) with respect to multiplication, type (3) with repect to subtraction> and type (4) with respect to division.> My point is that zero should bne trea as sui generis and recognized as> distinct from other numbers. (don't bother to tell me that zero is not an> operator)Yeah, but what does this have to do with infinite extension ininfinite dimensions or infinitely dense mass of infinite extension?Zero is fundamental because it is bound up with symetry, particularly> the> binary symetry of positive and negative numbers. Zero is paradoxical> because> it exists and does not exists or, rather, is a symbol for something> which is> impossible, non-existence, at least in the context of existence.You *might* have the beginnings of a point here. I don't know if you> have the osophical sophistication to turn this into a cogent> position. (My response would be that the context of existence isn't> the right context for an analysis of the number zero.)> I don't think that the term 'context of existence' is meaningful.> Mathematical systems exist, if only in thought, but they exist nonetheless.> The number zero cannot be analysed in the context of mathematics perse since> 'mathematics' is incapable of saying anything about its objects; that is the> role of metamathematics.> Russell's dictum is relevant here:> 'mathematics is the science in which we do not know what we are talking> about, and do not care what we say about it is true'.That's the spirit. :)Ideally, zero should be trea like infinity, as a necessary token for> a> number but not a proper number. Infinity is unreachable by any finite> operastion on infinite numbers. Zero is unreachable too by any finite> operation other than that of a-a. This last operation, then is rather> suspect and proves to be an annoyance in forms like II[1/(n-1) x 1/(n-2)> x> 1/(n-3) x ...]What the hell is a proper number?> A proper number is one which obeys all the operations of arithmetic without> exception, as a good logical object should. Unfortunately, zero is> recalcitrant with respect to division. Infinite and infinitesimal numbers> are improper and are kept out of the garden altogether.This seems ad hoc to me. If you look at the axioms for the PeanoArithmetic, you won't see an axiom for division. And it's obviousthat the natural numbers satisfy the PA axioms. I don't see why youfeel justified in calling division an arithmetic operation. By theway, the next section> And why isn't infinity a proper> number? Surely you're familiar with the logical results about the> existence of non-standard models for, say, the ordered Peano> Arithmetic which have objects which are greater than any other object.> Although infinity is referred to in analysis it is forbidden the status of> operand> in that context.I wasn't referring to analysis, but to results from mathematicallogic.> Infinity, as an object, is on the same logical status as any other> number.> It is not, for the reason given above. I agree that divergent series might> be said to have an infinite limit but this is meaningless unless one can say> which order of infinity it tends towards.The sentence you're replying to here is highly rela to the sentenceimmediately before it. I was not talking about limiting processeshere. I was talking about the existence of non-archimedean models ofordered Peano arithmetic. In such models, there are objects such thatthey are greater than any other object, including 1, 2, 3, ... . Thenatural interpretation for such an object would be as infinity, andin fact, similar ideas are used to develop the theory of non-standardanalysis, where objects called infinitesimals are given the samelogical status as numbers (so you can add a number and aninfinitesimal, say).> Infinity, as usually used in mathematics (like what you> described)> '(as you described it)' would be a neater expression for those who care> about good English.Well English, you mean. ;)> , is not an object, however. It is used as a sort of> short-hand for quantificational statements to describe the behavior of> the dependent variable (to set an upper bound, for instance) for any> given instance of the independent variable. Note that in the standard> definition of a limit, there is no mention of the word infinity. It> is all done in terms of existential quantification depending on> universal instantiation.> 'cid 'ooh> I know, that's why I said it wasn't a proper number or 'a number of the> usual sort' if you prefer.> I'm quite happy for infinite entites (not 'infinity' of course) to be> regarded as proper numbers. All you need are bigger 'premises'.But I'm not sure you realize that infinity can be a number, too. Infact, I'm sure you don't.'cid 'ooh=== === Subject: : Cracked World Crystals and Torn HologramsDear Friends :I forgot to tell Jack Sarfatti when I saw him in Santa Barbara about the importance of Clifford algebras in the evaluation of quantum bits at the Planck scale an all that ......in the calculation of the logarithmic corrections to the Bekenstein-Hawking black hole entropy.One uses Clifford Bits... The quanta OF spacetime are areas, volumes, hyper-volumes quantized in discrete Planck scale units. It is well known that Clifford algebras are very important in information theory.Jack: Right, so how do we formulate the t'Hooft-Susskind hologram in those terms.The quantum of 2D area is more fundamental than the quantum of 3D volume.Is it simply (Quantum of Volume) = (Quantum of Area)^3/2 ?They are not independent?Note, that in my theory the curved part of the Einstein metric field guv derives fundamentally from a quantum of area because the physical distortion of the spin foam lattice isdistortion = (Quantum of Area) (4-gradient of phase of local macro-quantum vacuum wave)guv is the symmetric strain tensor of this distortion first rank tensor.Clearly all the 4D information in guv is then scaled by the Quantum of Area.Since time is determined by relational changes in spacelike spin networks and the spin networks are basically determined by the areas the different pictures are intuitively similar.What happens in higher dimensions?In Cl(3) area is dual to line so that strings and area loops are dual images of each other.In my theory for the More is different emergence of c-number Einstein LOCAL gravity from the phase modulation of the giant local vacuum wave the loop gravity quantum of area is like the quantum of vortex circulation in Bohmian quantum theory whereIT FROM BITThis vortex defect picture is consistent with Hagen Kleinert's string defect picture of curvature and torsion as disclination and dislocation in the spin foam respectively. The defects are the Berry phase singularities in the LOCAL giant vacuum wave.All Kleinert world crystal lattices are spin foams.The quantum cosmology in Lee Smolin's book Three Roads to Quantum Gravity is all wrong IMHO because it over extrapolates micro-quantum theory and leaves out entirely More is different. All the Hartle-Gell-Mann stuff is wrong, the Fay Dowker stuff all wrong - or else I am wrong.Forget consistent histories. Forget Everett many worlds and all that. Linearity and unitarity are Mad Cows to the Slaughter House V!Now I really know what Wheeler means when he says:The Question is: What is The Question?The Achilles heel of consistent histories, causal sets and all that is what Wheeler is alluding to. More on that anon.Quantum Gravity is a House of Wild Cards, all Jokers on the shaky ground of signal locality, about to come tumbling down like Samson at The Temple.That Humpty Dumpty will not be put back together again, and The Emperor really has no clothes.http://stardrive.org/cartoon/MagicBean.html Extended Scale Relativity, p-loop Harmonic oscillator and Logarithmic Corrections to the Black Hole Entropy? Foundations of Physics vol. 33 Jack: Can you email me the files?Carlos: Notice that p-loops are closed strings, closed membranres, closed 3-branes....etc.... p-loops are the higher-dim generalization of ordinary loops.Jack: What about supersymmetry, the fermionic dimensions of non-commutative geometry?How does the detailed algebra of all this fit together?Carlos:ABSTRACT :A new relativity theory, or more concretely an extended relativity theory, actively developed by one of the authors incorporates 3 basic concepts. They are the old idea of Chew about bootstrapping, Nottale's scale relativity, and enlargement of the conventional time-space by inclusion of noncommutative Clifford manifolds where all p-branes are trea on equal footing. The latter allows one to write a master action functional. The resulting functional equation is simplified and applied to the p-loop oscillator. Its solution represents a generalization of the conventional quantum point oscillator. In addition, it exhibits some novel features: an emergence of two explicit scales delineating the asymptotic regimes (Planck scale region and a smooth region of a quantum point oscillator). In the most interesting Planck scale regime, the solution recovers in an elementary fashion some basic relations of string theory (including string tension quantization and string uncertainty relation). It is shown that the degeneracy of the first collective exci state of the p-loop oscillator yields not only the well-known Bekenstein-Hawking area-entropy linear relation but also the logarithmic corrections therein. In addition we obtain for any number of dimensions the Hawking temperature, the Schwarzschild radius, and the inequalities governing the area of a black hole formed in a fusion of two black holes. One of the interesting results is a demonstration that the evaporation of a black hole is limi by the upper bound on its temperature, the Planck temperature.Jack: I need to get to understand this...............................=== === Subject: : Re: Hilbert spaces in quantum mechanicsX-Cise: tanbanso@iinet.net.auX-CompuServe-Customer: YesX-Coriate: admin@interspeed.co.nzX-Ecrate: tanandtanlawyers.comX-Pose: george_cox@btinternet.comX-Punge: Micro$oftX-Sanguinate: themvsguy@email.comX-Terminate: SPA(GIS)X-Tinguish: Mark Griffith X-Treme: C&C,DWS>I had a course on quantum mechanics, but I'm having trouble relating>the maths to the physics.That's because there is a lot of hand waving in Physics. Some of whatthey do can be put on a rigorous footing and some can't.>We say that the set of state vectors for a system is a>Hilbert space.Yes. But, in fact, that's not enough. Google for rigged HilbertSpace for one approach to making things rigorous.>Is there a concrete realisation of this?Solutions to the wave equation. Except that they don't form a Hilbertspace, although they are dense in one.>is state space L2, the set of Lebesgue square->integrable functions R -> C ?Yes. But then you need to deal with issues of what domain an operatorhas.>But if so, this doesn't square with other concepts.Correct.>1) The momentum operator is not well defined:It is, but the domain is not the whole space.>i.e. the position operator has no eigenvectors.Correct.>3) Let < , > be the inner product on L2. It is a>bilinear map into C. How then can we possibly>write something like> =delta(x-x') ?Hand waving. Again, that can be cleaned up with a more sophisticaformulation, but I wouldn't expect that in an undergraduate or basicgraduate class.>4) =delta(x-x') seems to imply there are>uncountably many independent state vectors.>But a Hilbert space has a countable basis!Again, the problem goes away once you start using distributions andrigged Hilbert spaces.>I suppose that all these issues have a common stem.>It seems to me that L2 is altogether too large for>state space anyway; Both too large and too small.>Does quantum mechanics actually have rigorous mathematical>foundations? Yes, but it requires more machinery. QFT, OTOH, has problems.>How come textbooks always gloss over these>contradictions as if they all make sense?They're written by Physicists who don't care if the Mathematics areright. They're written for students who don't have the Mathematicalbackground to understand a more complica approach. there isn't timein the course to do it right and still cover the requisite Physics.That's the way that it has always been taught. There are probablyother reasons as well.-- Shmuel (Seymour J.) Metz, SysProg and JOATUnsolici bulk E-mail will be subject to legal action. I reservethe right to publicly post or ridicule any abusive E-mail.Reply to domain Patriot dot net user shmuel+news to contact me. Donot reply to spamtrap@library.lspace.org=== === Subject: : Re: Newbie Questions: SeriesThanks everyone for all the help!!Now, though, if I could get info on book / websites etc. on this typeof series (not just this, but more like table of series, if such athing exists), I will consider it as a happy bonus - I have asuspicion that I will find series-like problems many more times in myresearch (though my field is not math itself, but - for better or> En el mensaje:8447d48a.0401231624.93650f9@posting.google.com,> omniscient idiot escribi.97:> Dear all,I have two newbie questions. First, if one sums the following:> 1*k + 2*k^2 + 3*k^3 + ..... + (n-1)*k^(n-1) + n*k^n> with k = some constant number, what will the result be? Anyone has any> idea? Does this have a special name?> Without calculus, let> S = 1*k + 2*k^2 + 3*k^3 + ..... + (n-1)*k^(n-1) + n*k^n> kS = 1*k^2 + 2*k^3 + 3*k^4 + ..... + (n-1)*k^n + n*k^(n+1)> (1 - k)S = k + k^2 + k^3 + ... + k^n - n*k^(n+1)> = (k - k^(n+1))/(1 - k) - n*k^(n+1)> S = (k - k^(n+1))/(1 - k)^2 - n*k^(n+1)/(1 - k)=== === Subject: : Re: when NaturalNumbers = p-adics what alters in the Riemann Hypothesis Re: proof of the Riemann Hypothesis> (snips)> I would have had to consider two extreme cases: (1) that the RH is> completely false just as FLT is completely false when NaturalNumbers => p-adics. If taking that extreme route would have had me find out what> was flawed in my claimed two proofs below.> (2) the second extreme case is to say that something lies on the 1/2> Real line but not the NaturalNumbers of the illdefined notion of> finite-integers but rather instead the p-adics. Only the line is not a> straight line. And my second *alleged proof below* using a spiral sort> of touches or hints of a curved line.> (big snip)> TWO PROOFS OF THE RIEMANN HYPOTHESIS[clip]*Mr. Plutonium: Have you ever submit a paper for publication?earle*