for x,y > 7 twins(x+y) <= twins(x) + twins(y)
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From: spudnik on 14 Apr 2010 16:34 darn; I thought, from the header, you were using a multiplier of 7 ... and that made me realize, the professors who do that, are subverting the "big Oh" and "little oh" formalism. that partition of the triplet is so important, vuz Brun's constant! > for x,y > 7, twins(x+y) <= twins(x) + twins(y) > where twins is the prime twins counting function, > where 3,5,7 is considered as 2 twins. thus: just because it was British, I'd assume that the folks at E.Anglia did this, on purpose. "global" warming is almost & assiduously all computerized simulacra, and extremely limited reporting, about glaciers e.g. > >http://www.abc.net.au/unleashed/stories/s2868937.htm thus: to recap my reply to the TEDdies comments (as I am still listening to B.Greene's pop-sci talk ... zzzz), first of all, Minkowski made a silly slogan about ordinary phase-space, then he died. thank you! > http://www.ted.com/talks/lang/eng/brian_greene_on_string_theory.html thus: they were just at the library auditorium, selling the electromags to cure depression.... beats the heck out of electroconvulsing, but I missed the refreshments! thus: I didn't get the gist of the CBS reportage, although it seemed to be literate & wikipediaized (yeeha .-) seemed like "more decimal points," although there was a (wikip.) bibliographic note referring to Dicke -- I think, it was his paper that Einstein saw on one of his rare visits to his Caltech office, and pooh-poohed, regarding the predominant redshifitng of the heavens. thus: and, if at the centerof Sun is an iron core, the theory might have to be revized (don't laugh; not only was this a mainstream theory at one time, it may not have been laid to rest (in current research)). thus: Rob, you uneducated, global-warmed-over bog-creature -- did you create any oil, today?... seriously, that was amuzing about the cancellation-of-submission. reminds me of the time that Popular Science made an on-the-wayside attack upon S. Fred Singer; at the time they were owned by Times-Mirror, the then-owner of the LAtribcoTimes. the article was nominally and visually an aggrandizement of three professors (and taht could have included one of my own, at UCLA) of a theory about climate, which had been celebrated already (I think) with a Nobel. they included a mug-shot of the good doctor, along with no mention of his vitae; alas! thus: the Skeptics were a Greek cult in the Roman Pantheon, along with the Peripatetics, the Gnostics, the Solipsists etc. ad vomitorium; as long as the Emperor was the Top doG, you were left to your beliefs (til, of course, Jesus -- after it became the state church). thus: virtually all of "global" warming -- strictly a misnomer, along with Arrhenius 1896 "glasshouse gasses," except to first-order -- is computerized simulacra & very selective reporting, although a lot of the latter is just a generic lack of data (that is, historical data for almost all glaciers -- not near civilization). I say, from the few that I casually *am* familiar with, that *no* database shows "overall" warming -- not that the climate is not changing, rapidly, in the Anthropocene. thus: instead, we should blame Pascal for discovering, experimentally, his "plenum," which he thought was perfect. I mean, it's always good to have a French v. English dichotomy, with a German thrown-in for "triality." > of Newton's "action at a distance" of gravity, > via the re-adumbration of his dead-as- > a-doornail-or-Schroedinger's-cat corpuscle, > "the photon." well, and/or "the aether," > necessitated by "the vacuum." --Light: A History! http://21stcenturysciencetech.com --NASCAR rules on rotary engines! http://white-smoke.wetpaint.com
From: Transfer Principle on 14 Apr 2010 19:21 On Apr 14, 9:17 am, master1729 <tommy1...(a)gmail.com> wrote: > master1729 - Littlewood conjecture > for x,y > 7 > twins(x+y) <= twins(x) + twins(y) > where twins is the prime twins counting function where 3,5,7 is considered as 2 twins. In another thread, some standard theorists already explained why the "master-Littlewood conjecture" is likely false. In fact, Bau wrote yet another heuristic explaining why most standard theorists disbelieve this conjecture. I think that I have yet another analogy explaining the heuristic that argues against the conjecture. Let us define a sequence a_n as follows: a_0 = 2 a_1 = 3 a_2 = 5 a_3 = 7 a_4 = 11 So the sequence starts out looking like the primes. But for all subsequent values, we shall flip a coin. Then: a_(n+1) = a_n + 2, if the coin lands heads = 2a_n, if the coin lands tails. Let me simulate some coin flips right here: THTTTTHTHTHTTHHTHHHTHTTHHT Based on these coin flips, we have: a_5 = 22 a_6 = 24 a_7 = 48 a_8 = 96 a_9 = 192 a_10 = 384 a_11 = 386 a_12 = 772 a_13 = 774 a_14 = 1548 a_15 = 1550 a_16 = 3100 a_17 = 6200 a_18 = 6202 This sequence obviously increases exponentially faster than the primes does. Indeed, since the coin will land tails with probability 1/2, and we double after every tail, we see that a_n is approximately 2^(n/2). So it appears that we have a no-brainer: The Transfer Principle-Littlewood Conjecture: If given the sequence {a_n} above, we define a function: f_a(m) =def card({neN|a_n <= m}) then the conjecture states that: f_a(x+y) <= f_a(x) + f_a(y) Since a_n is approximately 2^(n/2), we have that f_a(m) is approximately 2log_2(m) (also written as 2lg(m)). But despite this, the Transfer Principle-Littlewood Conjecture is most likely false. For suppose after finding a_n for some n, we flipped heads five times in a row. Then we would have: a_(n+1) = a_n + 2 a_(n+2) = a_n + 4 a_(n+3) = a_n + 6 a_(n+4) = a_n + 8 a_(n+5) = a_n + 10 But if we let x=a_n and y=10 in the conjecture, then: f_a(a_n + 10) <= f_a(a_n) + f_a(10) n+5 <= n + 4 or 5<=4, a blatant contradiction. So we must conclude that as soon as we flip five heads in a row, the conjecture becomes false. But will we ever flip five straight heads? Well, after the sequence above, I eventually flipped five straight heads to give me the values: a_53 = 813681752 a_54 = 813681754 a_55 = 813681756 a_56 = 813681758 a_57 = 813681760 a_58 = 813681762 Thus x=813681752, y=10 is a counterexample to the conjecture. Of course, if we were to flip tails forever, or at least avoid flipping four straight heads, then our conjecture would be true. But this is extremely unlikely -- indeed, we will eventually flip five straight heads with probability _1_. Thus, the Transfer Principle-Littlewood Conjecture -- as well as the master1729-Littlewood Conjecture -- is akin to stating that we'll never flip five (or n for some large n) straight heads since each head has probability 1/2 (so n straight heads would have probability 1/2^n, a small number), but Bau's counterexample is akin to stating that if we flipped a coin _infinitely_ many times, then we'll eventually flip five (or n) straight heads with probability 1. In particular, it may be unlikely that all 48 of the numbers that Bau listed (n, n+2, n+6, n+8, etc.) are prime, perhaps as unlikely as flipping 48 straight heads, but just as we'll flip 48 straight heads eventually with probability 1, all 48 of the numbers will be prime for some n, with probability 1. Once again, I don't believe that tommy1729 should be ridiculed or called a "crank" for just stating his conjecture (unless one is prepared to call Hardy and Littlewood "cranks" as well). I had to reread Bau's post several times until I understood why conjectures of this type are probably false.
From: Transfer Principle on 14 Apr 2010 19:50 On Apr 14, 9:17 am, master1729 <tommy1...(a)gmail.com> wrote: > 1729 Actually, since I'm here in a tommy1729 thread, I might as well investigate the link he mentioned in a previous post: > http://sites.google.com/site/tommy1729/ At first I expected tommy1729 to explain more about the master 1729-Littlewood conjecture on that site, but instead I found the page called "debunking nonmeasurable set." Now we already know that tommy1729 is an opponent of AC, and AC is used in the usual ZFC proof of the existence of non-Lebesgue measurable sets (so that the proof would fail in a theory such as ZF+~AC). But on the page, I noticed what was written at the bottom: > x = 1/n lim n -> oo > Notice x IS NOT 0 , but an infinitesimal. And of course, we immediately see what's going on here. In standard analysis, there are no nonzero infinitesimals, and lim n->oo (1/n) is exactly zero in standard analysis. I've actually noticed this myself years ago. (I don't recall whether I ever mentioned this in any sci.math post.) The usual proof of a nonmeasurable set involves taking the unit interval and partitioning it into countably many sets, each of which would have the same measure. But in the (extended) standard reals, there is no real number r such that oo * r equals one (the measure of the unit interval). So we must say that these sets are nonmeasurable. But I often wonder, if we were allowed to assign nonstandard _infinitesimals_ as measures, then perhaps it may be possible to assign one to the measure of a Vitale set. This is related to another common so-called "crank" idea -- why should individual points have measure zero? Why can't we assign yet another infinitesimal (smaller than the infinitesimal for Vitale sets, of course) for the measure of each point? This idea is intuitive to many "cranks." The only set with measure zero would be the empty set. The measure would be completely additive -- that is, _uncountably_ additive as well as countably additive. Indeed, "cranks" often combine this with a nonstandard cardinality, so that the measure of a set would equal the product of its cardinality and some unit infinitesimal, the measure of a single point. (The measure of any set should be proportional to its cardinality.) Both RF and TO -- the infinitesimal "cranks," have proposed such a theory. MR, another infinitesimalist, has also looked at something similar. But of course, the standard theorists refuse to consider such ideas at all. Of course, probability theory would also work differently in such a theory, since probability is based on measure. For example, the probability I mentioned in my last post, of avoiding five straight heads when flipping a coin infinitely many times, would no longer be 1, but 1-(31/32)^oo, which is infinitesimally short of unity. And so I'd love to discover a theory in which the intutions of at least four "cranks" (RF, TO, MR, and tommy1729) would all be satisfied. Now some standard theorists might be wondering why I'm bending over backwards to accommodate tommy1729's idea of infinitesimal measure, right after attacking him for his insistence that his Littlewood conjecture is true. The answer is that I don't find a theory which redefines "prime," "twin prime," etc., to be as interesting as one in which sets can have nonzero infinitesimal measure. (The closest I can come to a theory in which tommy1729's conjecture is true is an ultrafinitist theory. For example, according to AP, 10^500 is the largest natural number. So according to AP, if a conjecture is true for all naturals at most 10^500, then it's true for all naturals. Since the estimated size of the counterexample mentioned in Bau's post is somewhat larger than 10^500, tommy1729's conjecture is probably true for AP-naturals.)
From: Jesse F. Hughes on 14 Apr 2010 20:10 Transfer Principle <lwalke3(a)lausd.net> writes: > Both RF and TO -- the infinitesimal "cranks," have proposed > such a theory. MR, another infinitesimalist, has also looked > at something similar. But of course, the standard theorists > refuse to consider such ideas at all. Neither Ross nor Tony has proposed any theory at all --- aside from Ross's nonsense about the wonders of the empty theory. Similarly, Mitch has not "looked at" anything at all, but merely occasionally makes pronouncements about how things "really" are near zero. On the other hand, the so-called standard theorists do not "object" to non-standard analysis. Perhaps the bulk of them choose not to work in NSA, but this is surely not what you mean when you write that they "refuse to consider" certain ideas. Why not give up the play-by-play analysis, since you have no talent for it? Just go ahead and suggest an interesting theory that you think approaches the ideas of Ross, Tony or Mitch. I don't promise that it'll get a whole lot of attention -- most mathematicians are not looking for unusual theories -- but I do wager that, assuming it's a formalizable first order theory, no one will object to it in the manner you pretend. -- Jesse F. Hughes "You people are the diminishment of a world." -- James S. Harris, to mathematicians.
From: master1729 on 15 Apr 2010 03:16
10^500. if we want to know the primes in an interval [10^500,10^500 + q] we need to sieve out all primes up to sqrt(10^500 + q). however the ' counterarguments ' only sieve out the primes up to about sqrt(q). thats their big mistake. surely lwalke you must have noticed that ?? tommy1729 the master |