April 2006 - mathfun@mailman.xmission.com

Re: [math-fun] batting averages
by Michael Reid 27 May '06

27 May '06

> SHOULDA > wrote: > At minimum, > (c74) block([keepfloat:true],linsolve((2*integer+1)/(5*integer+3)=0.3985)) > > (d74) [integer = 26.066] > > (c75) subst(ceiling(%),5*integer+3) > > (d75) 138 > Serves me right for not using continued fractions. > at bats. ok, that looks better. but how do you get the form (2n + 1) / (5n + 3) ? is there some way without continued fractions? btw, the c.f. method i described before can be cleaned up slightly. to find the "simplest" rational number in the closed interval [a, b] , compute the convergents for the cf's of a and b . suppose p/q and P/Q are the last two they have in common. then the next convergent for each will be of the form (p + Pn) / (q + Qn) for some positive integer n . if the simpler of these two is in the target interval, then that's the one. otherwise, replace n by n+1 in its expression as above, and that gives the simplest fraction in the interval. (this may require a and b to be positive, but it's easy to reduce to that case.) mike

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[math-fun] Problem Puzzler 1999
by Christian Boyer 10 May '06

10 May '06

Jean-Paul Delahaye publishes in the new issue of Pour La Science a very interesting article on Sudokus. He asked a very strange Sudoku problem WITHOUT any given numbers, only ">" and "<" signs between cells. He mentions that this problem was published in Puzzler, 1999, and that he does not know the solution. I have the solution, sent this morning to J.-P. Delahaye. Now looking for "Puzzler 1999" on Google, I find that Ed Pegg Jr. wrote an article in September 2005 including the same astonishing puzzle, figure 7 "Greater Than Sudoku" in: http://www.maa.org/editorial/mathgames/mathgames_09_05_05.html Perhaps that somebody else had already the solution? Ed, did you receive a solution to your article? Christian.

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[math-fun] Primitive, Hansen addition chains
by David Wilson 01 May '06

01 May '06

I know a lot of work has been done on minimal addition chains (shortest in number of additions). Presumably this is because minimal addition chains are of interest in optimizing certain operations, e.g, computing powers of large matrices or other complicated objects. I was curious if anyone has explored the idea of primitive addition chains, in which every element is necessary, that is, removing any element but the last does not yield an addition chain. It is easy to show that all minimal addition chains are primitive. But not all primitive chains are minimal (I state this without proof, I know it to be true, it requires only exhibiting a non-minimal primitive addition chain for some n). For n >= 4, the complete addition chain for n, (1,2,3,...,n), is not primitive, because the element n-1 may be removed yielding an addition chain. So what would be the smallest n for which some primitive addition chain for n is longer than a minimal chain for n? For small n, what is the length of the longest primitive addition chain for n? I would also like to know what Hansen chains are. The Mathworld description was not particularly enlightening to me, and there was very little information on the Web. An example with explanation might help me.

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[math-fun] a new identity for Zeta(5)
by simon.plouffe＠sympatico.ca 01 May '06

01 May '06

Hello, I have been working on those Zeta(2n+1) identities that involves infinite series with exp(-Pi*n) lately and I think I have found new formulas. They are interesting since they converge fairly rapidly, for example, that one for Zeta(5) : /infinity \ 5 | ----- | 694 Pi 6280 | \ 1 | Zeta(5) - ------- + ---- | ) --------------------| 204813 3251 | / 5 | | ----- n (exp(4 Pi n) - 1)| \ n = 1 / /infinity \ | ----- | 296 | \ 1 | - ---- | ) --------------------| 3251 | / 5 | | ----- n (exp(5 Pi n) - 1)| \ n = 1 / /infinity \ | ----- | 1073 | \ 1 | + ---- | ) ---------------------| 6502 | / 5 | | ----- n (exp(10 Pi n) - 1)| \ n = 1 / /infinity \ | ----- | 37 | \ 1 | - ---- | ) ---------------------| 6502 | / 5 | | ----- n (exp(20 Pi n) - 1)| \ n = 1 / It converges at a rate of 5.45 decimal digits per iteration. lprint(%): Zeta(5)-694/204813*Pi^5+6280/3251*sum(1/n^5/(exp(4*Pi*n)-1),n = 1 .. infinity)-\ 296/3251*sum(1/n^5/(exp(5*Pi*n)-1),n = 1 .. infinity)+1073/6502*sum(1/n^5/(exp( 10*Pi*n)-1),n = 1 .. infinity)-37/6502*sum(1/n^5/(exp(20*Pi*n)-1),n = 1 .. infinity): I have similar identities for Zeta(9) and Zeta(13) but not yet Zeta(3). As you can see there is some general patterns, other formulas can be found here : http://www.lacim.uqam.ca/~plouffe/ Simon Plouffe

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[math-fun] Re: favorite theorem
by James Propp 01 May '06

01 May '06

>>Let me say, only half in jest, that my favorite theorem in math is the >>classification theorem for finite sets, i.e., the fact that finite sets >>are classified up to isomorphism by the counting numbers (including zero)! > >And I'll respond, half-seriously: How would you prove that? I wouldn't call it a theorem in the theorem-proof sense. If a proof is defined (epistemo-sociologically) as a text that makes you believe and/or understand a proposition more than you did before you read the proof (leaving aside the question of what "believe" and "understand" mean), then I don't think this theorem *has* a proof! You can undoubtedly set up a formal system in which this claim is proved by induction, but nobody who didn't already accept the claim would be able to understand the formal system, let alone follow the proof. Instead, I am using the word "theorem" to mean any mathematical fact that makes you say "Well, will you look at that!" And this proposition is a real looker. We all encountered it when we were kids and noticed that a set doesn't change its cardinality when its elements are shifted around, and this agreeable invariance probably fuels many kids' enthusiasm for counting (even though their knowledge of this invariance is intuitive and most of them couldn't begin to articulate it as a general proposition). My serious point is that some mathematical facts may simultaneously be so profound and yet so commonplace that we don't recognize them as "theorems" as such. But I think I want to change my mind about my favorite mathematical fact. How about the fact that there are infinitely many counting numbers? The realization that you can keep on counting forever and never run out of numbers (even if you run out of names for them) was such a mind-blowing experience for my young self that I retained absolutely, positively NO memory of the experience. :-) But my mind remained permanently blown, so there was nothing for me to do with my life but become a mathematician... Is it a "theorem"? Well, I doubt that there are any formal bases for mathematics in which the axiom of infinity can be proved. Either you see it, or you don't. And once you've seen it, you can try to doubt it, but it's really hard to do. Jim

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Re: [math-fun] Re: favorite theorem
by Jpr2718＠aol.com 30 Apr '06

30 Apr '06

David Wilson wrote: > Yes and the proof is surprisingly simple. Let x = 1, y = 0. Works for noninteger D as well. Oops. Yes, of course. Thanks. I left out the word "nontrivial". Maybe I'll get it right this time: For D > 0, D an integer, not a square, x^2 - Dy^2 = 1 has a solution in integers x, y, with y > 0. John Robertson

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Re: [math-fun] a new identity for Zeta(5)
by dasimov＠earthlink.net 30 Apr '06

30 Apr '06

Simon Ramaplouffan writes: << 5 694 Pi Zeta(5) - ------- + . . . 204813 >> This looks like a very cool identity, but can we assume that the first symbol after Zeta[5] is an equals sign rather than a minus sign? --Dan

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Re: [math-fun] Re: favorite theorem
by Jpr2718＠aol.com 29 Apr '06

29 Apr '06

> On Fri, 28 Apr 2006, Steve Gray wrote: >> ... On these scores, e^(i pi)+1=0 is unbeatable. > I am sure that I am missing here something. What is > the definition of e^(x+iy) ? Isn't it > e^(x+iy)=e^x (cos y + i sin y) ? > Emeric Hmm. It's been a long time, but my recollection is that the definition of e^z for z complex is that it is the (necessarily unique) analytic continuation of e^x for x real. Another great theorem is that every prime congruent to 1 modulo 4 is a sum of squares in an essentially unique way. But my favorite is that for D > 0, D an integer, not a square, x^2 - Dy^2 = 1 has a solution in integers. John Robertson

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[math-fun] Dimension of real division algebras
by dasimov＠earthlink.net 29 Apr '06

29 Apr '06

Gene wrote: << --- dasimov(a)earthlink.net wrote: ... > * Real division algebras occur only in dimensions 1,2,4,8. ... You mean to say: Finite dimensional real division algebras. There are lots of infinite dimensional real division algebras. Some examples are the fields of rational functions R(x), R(x,y), etc. >> Thanks for pointing that out -- it hadn't occurred to me. (Plus, I seem to recall JHC's stating in this venue that there exist no infinite-dimensional real division algebras.) But I see no problem with R(x), etc., as real division algebras. Which led me to wonder what a basis might be for R(x) as a real vector space. I suppose one possible basis for R(x) is { x^k / P(x)^n : P(x) a real monic poly. irreducible* over R with k, n nonnegative}. *I.e., P(x) is of form x-c (c real), or else x^2 + bx + c (b,c real with b^2 < 4c). This implies R(x) has dimension 2^aleph_0 as a vector space over R. Could JHC have been referring only to the nonexistence of real division algebras of dimension aleph_0 ? --Dan

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Re: [math-fun] Re: favorite theorem
by dasimov＠earthlink.net 29 Apr '06

29 Apr '06

Emeric wrote: << On Fri, 28 Apr 2006, Steve Gray wrote: > ... On these scores, e^(i pi)+1=0 is unbeatable. I am sure that I am missing here something. What is the definition of e^(x+iy) ? Isn't it e^(x+iy)=e^x (cos y + i sin y) ? >> Of course not: e^x began life as the inverse of tne integral from 1 to x > 0 of 1/x; sine and cosine began their lives as projections of a point moving around the circle at constant speed. Then these were extended in the unique natural way to the complex numbers. Then voila! -- an unexpected connection is made. --Dan

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