January 2004 - mathfun@mailman.xmission.com

[math-fun] Fwd: conjecture => RH
by Richard Schroeppel 22 Jan '04

22 Jan '04

The message below was sent to the old Arizona math-fun list. (The first non-spam message in months.) Mr Agol is not a Math-Fun subscriber, so CC him on any comments. Rich rcs(a)cs.arizona.edu ---------------------- Date: Wed, 21 Jan 2004 23:53:43 -0600 (CST) From: Ian Agol <agol(a)math.uic.edu> To: math-fun(a)CS.Arizona.EDU Subject: conjecture => RH If one writes: 1/Zeta[1/(1-z)]= Sum a_n z^n, then RH is equivalent to limsup |a_n|^(1/n) = 1. Computing a_n with mathematica, it seems that |a_n|<1, and oscillates between +-.02 to n=900. It seems reasonable to conjecture that a_n -> 0. Does anyone know if this conjecture has been made before, or if it is implied by some stronger published conjecture? Can anyone compute more terms of this series to see if it holds up? Using known bounds on the Stieltjes constants, I can numerically show that the radius of convergence of the series is at least .56. Can anyone improve on this? thanks, Ian Agol

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Re: [math-fun] Simple Question I think
by asimovd＠aol.com 22 Jan '04

22 Jan '04

David Wilson asks: << Is every rational point on the plane equidistant from two integer points? >> Here's a question in this vein that I think is unsolved: In the plane, given a unit square, is there any point whose distance to each corner is rational? Dan -- NOTE: Please direct any responses to <asimov(a)msri.org>, since I may not keep this AOL e-address much longer -- thanks.

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[math-fun] Re: Egyptian-Fraction Expansions Of REALS
by Richard Guy 22 Jan '04

22 Jan '04

I haven't been paying much attention to this, so may be barking up a different tree. Has attention been given to expressing a positive real as the sum of distinct egyptian fractions less than one? For example pi = {2,3,4,...,33,34,43,7884,99771155,16909840222780252,...} (or thereabouts) Of course, we all know why unit fractions are called `egyptian fractions' -- from the Hungarian `egy' for one. R. On Wed, 21 Jan 2004, Jud McCranie wrote: > At 07:18 PM 1/21/2004, Leroy Quet wrote: > For instance, how would the EF-expansion of pi (or pi-3) compare with > >that of 1/pi? > > I expect it to be completely different. > >

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[math-fun] Simple Question I think
by David Wilson 22 Jan '04

22 Jan '04

Is every rational point on the plane equidistant from two integer points?

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[math-fun] counting groups
by Richard Schroeppel 21 Jan '04

21 Jan '04

JHC's note suggested a couple of questions: What's the smallest ODD group-abundant number? (Ed, your table of G(N) only went up to 799.) The smallest odd abundant number is 945; we might guess the smallest odd-g-a is 2187 or 6561. I wouldn't be quite so sure that there are no G-perfect numbers. I'm curious when G(N)=1. This would be the cyclic group ZN, or (mod N). Since A|B -> G(A)<=G(B), and G(P^2)=2 (ZPxZP and (mod P^2)), N must be square-free. Moreover, G(PQ) = 2 when P|Q-1, and 1 otherwise, so we get an extra condition on the primes dividing N: PQ|N -> not(P|Q-1). Is this set of conditions sufficient? The smallest three-factor N meeting these conditions is 255 = 3.5.17, and, indeed, the table shows G(255)=1. The density of such N is <=epsilon. If I've got the predicate right, then the density eventually exceeds any ((loglogN)^K)/logN. Sparse, but not that sparse. More generally, when is G(MN)=G(M)*G(N)? If gcd(M,N)=1, we expect at least >= just from the cross products, after we've proved they are all distinct. Groupies? Rich rcs(a)cs.arizona.edu

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[math-fun] Tenuous Associations
by David Wilson 21 Jan '04

21 Jan '04

I have in front of me an article "The Mystery of Time" from "The Mongol Messenger", a Mongolian newspaper out of Ulaan Bataar, dated January 2002. This article was sent to me by a friend in the country, who thought I might find it of mathematical interest. The content of the article is quite entertaining, though of little mathematical merit. The author is apparently from the UK. I would consider the article a crackpot piece, though it appears to be well written and of a less confrontational tenor than many such articles I have read. It contains some interesting equations(sic). It is difficult to summarize the article, which seems to be a hodgepodge of mystical and mathematical associations revolving around the concept of time, but with no overriding premise or point. I have a mosaic scan of the article available upon request, about 6MB. The author seems to be fixated on certain historical dates, 1543 (Copernicus's heliocentric theory), 1618 (Kepler's laws), 1687 (Newton's Principia), 2001 (unified physics?), and 2060 (Newton's Last Judgment date?). The Hindu cycle (4320000000 years?) also figures prominent. All sorts of wondrous mathematical associations are used to justify or explain the significance of these dates. The types of things done in this article are common in crackpot literature, particularly the misuse of associations. Since mathematics is so rich in associations, and numbers are so pervasive in our affairs, its is only natural that unjustifiable mathematical associations between unrelated numbers would suggest themselves. And if you push it, you can find "significant" relationships between all sorts of unrelated numbers. I was wondering if there is a name for the practice of drawing tenuous assocations between unassociated entities from a rich association domain, and specifically, the practice of drawing mathematical associations between mathematically unrelated numbers? For instance, let us suppose I see special historical significance in the person of George Washington (which I do). I notice that his birth year, 1732 is reminiscent of sqrt(3) = 1.732+, and from that propose some sort of mystic significance. So let us drag down this mystic association. First, we assume that George Washington's birth date is a significant determinant or indicator of his personal signficance, not an uncommon practice, certainly a major underpinning of astrology and other mystical arts. Then we assign his birth year the number 1732. This is a customary, but ultimately arbitrary number. It indicates that the Earth has orbited the sun somewhere between 1732 and 1733 times between the unknown and probably incorrect time of the birth of Jesus Christ and the time of birth of George Washington. 1732 has a visual similarity to the number 1.732, an approximation to the square root of 3. The visual similarity rests on the peculiarities of our decimal notation, which itself rests on the number 10, presumably the number of fingers on our two hands. The actual mathematical association is 1732 = floor(1000 * sqrt(3)). I could take this exploration further, but suffice it to say that this obvious mystic significance I have assigned to George Washington is based on a not-so-simple mathematical relationship between several unrelated quantities, including the time of George Washington's birth, the time of Jesus Christ's birth, errors in the calculation of the latter date, the period of revolution of the Earth about the Sun, the number of fingers on the hands of human beings, and the arbitrary integers 3 and 1000. To which of these quantities does the relation 1732 = floor(1000 * sqrt(3)) lend significance? And certainly there are many other birth dates and events occuring in the year 1732 as we reckon it, which were not nearly so historically significant as the birth of George Washington, which leads us to question the force of the association. Is there a name for the inappropriate use of tenuous associations such as the one described above? If not there should be. Because numbers are rife with mathematical associations, and are so pervasive in our everyday experience, they lend themselves to this sort of abuse, and there should be a specific term for this type of abuse of numerical associations.

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[math-fun] Frost Ration (haiku)
by wouter meeussen 21 Jan '04

21 Jan '04

Frost Ration, or Blue Screen of Death. Frost in early june, Micro Ice to pierce Soft fruit; this Screen is Frozen, please Reboot. W.

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[math-fun] self-generating continued fractions
by Kimberling, Clark 21 Jan '04

21 Jan '04

Suppose r>0. Let's mimic the standard algorithm for finding the simple continued fraction of r, but, instead of taking successive numerators =1, take each to be the most recently generated "a(k)" - like this: r(0)=r, a(0)=[r(0)] r(1)=a(0)/(r(0)-a(0)) a(1)=[r(1)] r(2)=a(1)/(r(1)-a(1)) a(2)=[r(2)], and so on. Call the sequence <a(0),a(1),a(2),...> the self-generating continued fraction of r, and denote it by S(r). Has anyone encountered this previously? One very attractive example has been known for a long time: 1/(e-2) = <1,2,3,4,5,6,7,8,...>. An easy problem is to determine all r for which S(r) is a constant sequence. Probably more interesting, though, are examples like these: <1,2,4,8,16,32,64,...> = 1.4086159797... <1,3,5,7,9,11,13,...> = 1.2831923416... <1,4,7,10,13,16,...> = 1.221107010123... Are these numbers "known"? Let R_n denote the set of r corresponding to the arithmetic sequences of positive integers congruent to k mod n for k=1,2,...,n. What can be said about the numbers in R_n? Best wishes, Clark Kimberling

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Re: [math-fun] When liberals do science...
by Jeffrey Shallit 21 Jan '04

21 Jan '04

> From: Eugene Salamin <gene_salamin(a)yahoo.com> > [snip] > Reference: Petr Beckmann, "The Health Hazards of Not Going Nuclear", > Golem Press. > > Gene I really wish certain people could restrain themselves from promulgating their political views on this list. I subscribe because it occasionally has interesting math, but I'll unsubscribe in an instant if this degenerates into mindless attacks on liberals or conservatives. By the way, wasn't Petr Beckmann the crackpot who claimed relativity was a hoax? Jeffrey Shallit

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[math-fun] claim of proof for No Odd Perfect Numbers
by Richard Schroeppel 21 Jan '04

21 Jan '04

Paper (*cross-listing*): hep-th/0401052 From: Simon Davis Dr <davis(a)math.uni-potsdam.de> Date: Thu, 8 Jan 2004 23:56:42 GMT (9kb) Title: A Proof of the Odd Perfect Number Conjecture Authors: Simon Davis Comments: TeX, 13 pages Report-no: RFSC 04-01 Subj-class: High Energy Physics - Theory; Number Theory The finiteness of the number of solutions to the rationality condition for the existence of odd perfect numbers is deduced from the prime decomposition of the product of repunits defined by the sum of the divisors, $\sigma(N)$. All of the solutions to this rationality condition satisfy the inequality ${{\sigma(N)}\over N}\ne 2$. This technique is then used to demonstrate that there are no odd perfect numbers $N=(4k+1)^{4m+1} \prod_{i=1}^\ell q_i^{2\alpha_i}$. http://arXiv.org/abs/hep-th/0401052 13 pages ---------------------- I looked at this, but haven't puzzled through it yet. A crank? Putting the paper in hep-th is curious, and the author affiliation is unusual. The surface markers look mostly reasonable: real references, real notation. The logic is abbreviated, and it's hard to tell if it's merely explanatory. Page 1 leaps immediately into complicated algebra, eschewing background. There are well known required shapes for hypothetical odd perfect numbers. (These arise from equating 2N with the formula for the sum of divisors of N, and analyzing the even/odd properties of base P repunits. "Elementary" number theory.) One requirement is that all but one prime divisor have even exponent, making a square of a prime or prime power. One part of Davis's argument is that the divisor sum term for these primes (a factor of P^2 leads to a divisor-sum factor of P^2 + P + 1, a base P repunit 111) leads to new primes > P, which must then appear as new divisors of the hypothetical perfect number. For example, 7^2 -> 7^2 + 7 + 1 = 57 = 3 * 19 giving the new prime 19. Continuing the argument, 19^2 -> 381 = 3 * 127, and so on. If this always led to new larger primes, it would prove NoOPN. But it doesn't always give new primes (67^2 -> 4557 = 3 * 7 * 7 * 31 and 79^2 -> 6321 = 3 * 7 * 7 * 43), and I think his argument is concerned with handling the exceptional cases. Rich rcs(a)cs.arizona.edu

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