January 2005 - mathfun@mailman.xmission.com

[math-fun] Re: Freeman Dyson responds...
by Gary McGuire 06 Jan '05

06 Jan '05

>[math-fun] Freeman Dyson responds to the question "What do > you think is true even though you cannot prove it"? >... >order. Now my statement is: it never happens that the reverse >of a power of two is a power of five. >... I find Dyson's response interesting. He claims that he has a statement that is true and unprovable. However, he only explains why he *believes* that the statement is true. This is different from Godel, who has statements that he *knows* to be true. Dyson's arguments that the statement is 1) true, and 2) unprovable, are heuristic. Perhaps the arguments can be made rigorous? He says that "The digits in a big power of two seem to occur in a random way without any regular pattern." This is the basis for the whole thing. Is there a proof of this? I don't see why this is necessarily true, although I can't think of a reason why not. I would naively have thought that the first digit in the primes was uniformly distributed, because I can't think of a reason why there are more primes beginning with a 1 than with a 9, but this is not true. (Anyone got a heuristic argument for this?) Gary McGuire

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[math-fun] Modular square root
by David Wilson 06 Jan '05

06 Jan '05

Is there a fairly fast algorithm to find the square root of a number to a given prime modulus p?

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[math-fun] Re: Simple(?) Fib problem
by David C Terr 06 Jan '05

06 Jan '05

This is an interesting problem. If m is a multiple of 8, then the Fibs do not have every residue mod m since no Fib is congruent to 4 mod 8. Otherwise, I need to think about it further. Dave "David Wilson" <davidwwilson(a)comcast.net> 01/05/2005 07:17 AM To: "Sequence Fans" <seqfan(a)ext.jussieu.fr>, "math-fun" <math-fun(a)mailman.xmission.com> cc: Subject: Simple(?) Fib problem For which m for do the Fibs have every residue mod m?

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[math-fun] Re: Simple(?) Fib problem
by Jud McCranie 05 Jan '05

05 Jan '05

>"David Wilson" <davidwwilson(a)comcast.net> > >For which m for do the Fibs have every residue mod m? A quick test shows that these do: 1,2,3,4,5,6,7,9,10,14,15,20,25,27,30,35,45,50,70,75,81,100,125,135,150,175 ... but this doesn't necessarily rule out the others.

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[math-fun] Learning computer programming for a bright 13-year old ?
by Daniel Asimov 05 Jan '05

05 Jan '05

My very bright 13-year-old nephew would like to learn computer programming (from scratch). He's quite familiar with using software, especially having used Midi music software for several years now. At this point I don't think he has any specific goal of building a particular kind of software; he just wants to learn what computer programming is all about. Has anyone had success with a particular language and/or book in similar circumstances? All suggestions are welcome. At this point I'd like to give him something simple (so probably Java is out) but not too simple (so maybe Alice is out). But my mind's not made up about these, either. Many thanks for your ideas. --Dan

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[math-fun] Pancake cutting problem
by lkmitch＠att.net 04 Jan '05

04 Jan '05

I was looking at the "pancake cutting" problem on MathWorld (more precisely, the problem of circle division by lines): http://mathworld.wolfram.com/CircleDivisionbyLines.html and its cousin, the problem of the plane divided by circles: http://mathworld.wolfram.com/PlaneDivisionbyCircles.html and I'm wondering if there are "nice" ways to draw the images for larger numbers of cuts. I realize that "nice" is horribly ill-defined, but I'm thinking of images that are aesthetically appealing, as well as mathematically accurate. Or, if anyone has any hints to offer about drawing my own, I'd be oh so appreciative. Thanks, Kerry Mitchell

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[math-fun] Freeman Dyson responds to the question "What do you think is true even though you cannot prove it"?
by Thane Plambeck 04 Jan '05

04 Jan '05

I thought this was interesting ---snip---- Since I am a mathematician, I give a precise answer to this question. Thanks to Kurt GÃ¶del, we know that there are true mathematical statements that cannot be proved. But I want a little more than this. I want a statement that is true, unprovable, and simple enough to be understood by people who are not mathematicians. Here it is. Numbers that are exact powers of two are 2, 4, 8, 16, 32, 64, 128 and so on. Numbers that are exact powers of five are 5, 25, 125, 625 and so on. Given any number such as 131072 (which happens to be a power of two), the reverse of it is 270131, with the same digits taken in the opposite order. Now my statement is: it never happens that the reverse of a power of two is a power of five. The digits in a big power of two seem to occur in a random way without any regular pattern. If it ever happened that the reverse of a power of two was a power of five, this would be an unlikely accident, and the chance of it happening grows rapidly smaller as the numbers grow bigger. If we assume that the digits occur at random, then the chance of the accident happening for any power of two greater than a billion is less than one in a billion. It is easy to check that it does not happen for powers of two smaller than a billion. So the chance that it ever happens at all is less than one in a billion. That is why I believe the statement is true. But the assumption that digits in a big power of two occur at random also implies that the statement is unprovable. Any proof of the statement would have to be based on some non-random property of the digits. The assumption of randomness means that the statement is true just because the odds are in its favor. It cannot be proved because there is no deep mathematical reason why it has to be true. (Note for experts: this argument does not work if we use powers of three instead of powers of five. In that case the statement is easy to prove because the reverse of a number divisible by three is also divisible by three. Divisibility by three happens to be a non-random property of the digits). It is easy to find other examples of statements that are likely to be true but unprovable. The essential trick is to find an infinite sequence of events, each of which might happen by accident, but with a small total probability for even one of them happening. Then the statement that none of the events ever happens is probably true but cannot be proved. ---snip---- more here: http://www.edge.org/q2005/q05_print.html -- Thane Plambeck http://www.plambeck.org/ehome.htm

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[math-fun] 2005
by Marc LeBrun 03 Jan '05

03 Jan '05

Happy MMV! From the OEIS we learn, among many other things, that 2005 is... Vertically symmetric [A053701] 6 4^5 + 5 4^4 + 4 4^3 + 3 4^2 + 2 4^1 + 1 4^0 [A059045] The value of 11 n^2 + 11 n + 3 for n=13 [A006222] The value of (14 n^3 - 21 n^2 + 13 n)/4 for n=10 [A071229] The value of ((2n - 1) 3^n + 1)/4 for n=6 [A014915] The coefficient of x^14 in (1-x)/(1-x-x^2-x^3-x^4+x^5) [A033305] Pick integers x, y and z between 1 and 32 inclusive. The expression xy+yz+zx can take on any one of exactly 2005 distinct values [A100440, recently contributed on 21 Nov 04]. Expand (p+1)^8-1 and then "umbrally" replace p^n by the n-th prime. The result sums to 2005 [A050513] Perhaps the most interesting and apropos is the following variant of the well-known "birthday paradox": Pick 53 years at random since the birth of Christ (ie any AD) and the chances are just better than half that two will coincide, but pick only 52 and the odds are just less than even. [A072829, if I've interpreted it aright] Hope you find these factoids of interest. Best wishes for the New Year! --Marc LeBrun

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