April 2012 - mathfun@mailman.xmission.com

[math-fun] > 4. Re: n and n^3 have all digits odd (Tom Duff)
by Tom Duff 20 Apr '12

20 Apr '12

>I'm not sure I understand; Me neither. I have no idea what I was thinking. -- Tom Duff. There's only one power stronger than direct marketing.

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[math-fun] Dark matter + gravity mysteries. Something is very wrong.
by Warren Smith 20 Apr '12

20 Apr '12

Yes, dark matter does couple gravitationally with normal matter, hence can lose energy. Also, it might couple via the "weak force" which perhaps could allow it to be detected. Trouble is, both the weak force and gravity are very weak interactions so such losses are very small and low rate.. ----- I think that "Moffat gravity" and "Bekenstein gravity" would explain the latest local measurements which find no dark matter in our vicinity (they are rather ad hoc theories of gravity a good deal more complicated than Einstein/Newton but whose goal is to explain the dark matter gravitational effects without needing dark matter to exist). The observational paper, by the way, is C. Moni Bidin, G. Carraro, R.A. Mendez, R. Smith: Kinematical and chemical vertical structure of the Galactic thick disk II. A lack of dark matter in the solar neighborhood, http://arxiv.org/abs/1204.3924 17 April 2012, accepted Astrophysical Journal. "We estimated the dynamical surface mass density Sigma at the solar position between Z=1.5 and 4 kpc from the Galactic plane, as inferred from the kinematics of thick disk stars. The formulation is exact within the limit of validity of a few basic assumptions. The resulting trend of Sigma(Z) matches the expectations of visible mass alone, and no dark component is required to account for the observations. We extrapolate a dark matter (DM) density in the solar neighborhood of 0+-1 M_sun/pc^3, and all the current models of a spherical DM halo are excluded at a confidence level higher than 4sigma..." However there is another paper I.D. Karachentsev: Missing Dark Matter in the Local Universe, Astrophysical Bulletin 67,2 (2012) 123-134 http://arxiv.org/abs/1204.3377 "A sample of 11 thousand galaxies with radial velocities < 3500 km/s is used to study the features of the local distribution of luminous (stellar) and dark matter within a sphere of radius of around 50 Mpc around us... missing dark matter." which it seems to me devastates ALL THREE: Einstein+Dark matter, Moffat, and Bekenstein gravity -- if this is believed it seems to me all 3 theories of gravity must be false. And here is a somewhat ranty 50 page review paper: Pavel Kroupa: The dark matter crisis: falsification of the current standard model of cosmology http://arxiv.org/abs/1204.2546 which seems to think Einstein gravity + dark matter is clearly busted. So it seems to me that gravity must be much more complicated than (almost) everybody thought, OR these observationists are making huge mistakes. I'd like to know which.

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[math-fun] n and n^3 with all odd digits
by Warren Smith 20 Apr '12

20 Apr '12

If we consider not base 10 but rather base 100, 1000, etc, then the requirement that n^3 have all odd digits using base 10^K is equivalent to every Kth digit being odd back in normal base 10. This is a different "test set" than Victor Miller was testing by asking that n^3 mod 10^k have all k digits odd; this asks for the least-sig k digits to be odd. Similarly by asking about 10*n^3 in base 10^K we could get a still different "test set" etc. --- Another thing one could consider is "carryless" problem, that is, consider not numbers but rather polynomials in a dummy variable with integer coefficients. Then ask about whether both P(x) and P(x)^3 can have all coeffs odd. This can also be done for formal power series instead of polynomials. Can we find such a series, all of whose infinite number of terms are odd, and its cube ditto? Answer=NO because (a0+a1*x+a2*x^2+a3*x^3+...)^3 = a0^3 +3*a0^2*a1*x +(2*a0^2*a2+a0*a1^2+2*a1^2*a0+a2*a0^2)*x^2 +... and the coefficient of x^2 is always even.

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Re: [math-fun] n and n^3 have all digits odd
by Tom Duff 20 Apr '12

20 Apr '12

Quoth rcs(a)xmission.com > Cubes (mod 9) are always 0, 1, or 8. > The low order ternary digits are 00, 01, or 22. --Rich To settle the decimal case, you have to go up to mod 10^9 at which point you will discover that none of the 1000 possibilities is all odd digits. I did it by grepping the output of this program: #include <stdio.h> int main(){ for(int i=0;i!=1000;i++) printf("%09d\n", i*i*i); return 0; } -- Tom Duff. Seize you your paper and go into an art seminar!

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[math-fun] Cheshire Cat Dark Matter
by Warren Smith 19 Apr '12

19 Apr '12

re Henry Baker's large fantasy collection... 1. yes, gravitational lenses work for gravity as well as photons... a fact which nobody has ever had any interest in before (and I don't see why they should now either) 2. dark matter is an example of stuff which never "coalesces" particularly. Or at least that was the usual picture: DM consists of some kind of massive fundamental particles which have little or no interaction with normal matter particles or each other, besides gravitatonal. They got generated in the big bang at some temperature, and have gradually cooled off ever since. There is a blob of dark matter forming a rough spherical blob centered at each galaxy. The galaxies are however not sphere shaped usually. The rotation curves of galaxies are way different from what they should have been which is the main way they deduced the DM is there. Normal matter hits itself and emits heat hence slows down and coalesces. Dark matter never hits, never emits heat, never slows down, never coalesces particularly. 3. there were some crazy non-Einstein theories of gravity, some of which were claimed to explain the gravitational effects of dark matter without needing to assume the existence of the dark matter. And perhaps they indeed will need to be dusted off to explain this discrepancy. You've inspired me to send a note to John Moffat, inventor of one non-Einstein gravity theory, to bring this to his attention. -- Warren D. Smith http://RangeVoting.org

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[math-fun] odd digits in n and n^3
by Warren Smith 19 Apr '12

19 Apr '12

> Certainly we can prove this theorem for base 2, can't we? How many cubes > are there of the form 2^n-1? Something is telling me, "one," but I can't > prove it. --Yes, Catalan's conjecture, nowadays theorem, says A^n - B^m = 1 has only one solution 3^2 - 2^3 = 9-8 = 1 if A,B,n,m, all >=2. Hence 2^n-1 is never a cube if n>=2.

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[math-fun] Archive of Research Papers
by David Makin 19 Apr '12

19 Apr '12

A friend (Samuel Monnier) posted this link on Facebook, I thought it may be of interest.... http://arxaliv.org/ The meaning and purpose of life is to give life purpose and meaning. The instigation of violence indicates a lack of spirituality.

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[math-fun] n and n^3 have all digits odd
by Warren Smith 19 Apr '12

19 Apr '12

> From: Victor Miller <victorsmiller(a)gmail.com> > To: math-fun <math-fun(a)mailman.xmission.com> > Subject: Re: [math-fun] n and n^3 have all digits odd > Message-ID: > <CAEe1e0zHEW=rufvv1R7MmW+kGY-54Und+MJhsuoZF+NuzBxw7w(a)mail.gmail.com> > Content-Type: text/plain; charset=ISO-8859-1 > > I'm doing some calculations pertaining to this. For all positive integers > k define the set > > S_k := { 1 <= n < 10^k | OddDigits(n) & OddDigits((n^3) mod 10^k) } > > It looks like (log # S_k)/k --> 1 (where log is the natural log). Notice > that 10^k-1 is always in S_k. --that "e" should come up here seems quite amazing. But anyhow, let me make a simple remark. Let T_k = {1 <= n < 10^k | OddDigits(n) } and obviously #T_k = 5^k. So assuming MIller correct, (#S_k)/(#T_k) behaves for large k approximately like (e/5)^k where e=2.71828... So now, the chance that all 3k digits of n^3 (it is about 3k digits long) all are odd, is presumably (e/5)^(3k). Hence, the total cardinality of the original problem's set R_k = {1 <= n < 10^k | OddDigits(n) & OddDigits(n^3)} is presumably about #R_k =approx= 5^k * (e/5)^(3k) = A^k where A = e^3 / 5^2 = 0.8034. This compares with Charles Greathouse's previously-posted estimate #R_k =approx= (5/8)^k where 5/8 = 0.625. Either way, the conclusion is the same -- the total cardinality of R_1 U R_2 U R_3 U ... is presumably finite - but it's now less clear than it appeared from Greathouse's original estimate.

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[math-fun] interstellar microbes?
by Warren Smith 18 Apr '12

18 Apr '12

Another ingredient to add to the stew is this: http://www.cfa.harvard.edu/news/2012/pr201212.html There are supposedly a number of "rogue planets" greater than the number of stars, which were ejected from their systems. These scientists claim based on "simulations" that the rogues can be captured by other stellar systems, and 3-6% of stars would thus-capture a rogue. "Stars trade planets just like baseball teams trade players," said Hagai Perets of the Harvard-Smithsonian Center for Astrophysics. This (a) gives another mechanism for panspermia and (b) suggests that there could be some serious risks for Earth... assuming their claims are not crazy... I think, however, there is considerable likelihood their claims are crazy. Why? If they were correct then since the number of stars and rogues were assumed in their sim to be exactly equal, we would conclude that stellar "near miss" swingbys at distances well within the solar system, would occur for well over 3-6% of all stars. I.e. about 100%. That in turn would mean that solar systems basically could not exist. But we know many exist and they are suspected to be quite common.. How can this contradiction be resolved? The press release says their sim focused not on Joe Average Star, but rather stars in "young clusters." Oh. It did not give a link to actual paper, so one's left kind of guessing. -- Warren D. Smith http://RangeVoting.org

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[math-fun] planets orbiting binary stars
by Warren Smith 18 Apr '12

18 Apr '12

http://www.cfa.harvard.edu/news/2012/pr201202.html there are currently 3 binary star-systems known which each have at least 1 planet. Presumably millions exist.

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