April 2008 - mathfun@mailman.xmission.com

[math-fun] Generalized "square pyramid" question
by Dan Asimov 19 Apr '08

19 Apr '08

Since 1^2 + 2^2 + 3^2 + ... + n^2 = Q^2 has only one nontrivial solution, I wonder what the prognosis is for exponents k higher than 2: QUESTION: For which k does there exist any nontrivial solution (n,Q) to 1^k + 2^k + 3^k + ... + n^k = Q^k ??? In particular, are there any solutions at all for k > 2 ? If so, what is the cardinality of the set of k for which solutions exist? (In particular, is it finite?) --Dan _____________________________________________________________________ "It don't mean a thing if it ain't got that certain je ne sais quoi." --Peter Schickele

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Re: [math-fun] a perplexing asymmetry
by Steve Witham 17 Apr '08

17 Apr '08

>From: James Propp <jpropp(a)cs.uml.edu> (me:) > >It looks sort of log like, but if you graph those numbers above, >>there are very piecewise-linear features, which doesn't make me >>confident I know what it's doing. > >Can you post the graph someplace so I can see what you mean? http://www.tiac.net/~sw/2008/04/RounderRouter/wandergraph.jpg --Steve

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[math-fun] Rounder-Router goodies
by Steve Witham 17 Apr '08

17 Apr '08

"Where the imaginary will take us this time?" --Ken Nordine I put all my rounder-router stuff on a web page: http://www.tiac.net/~sw/2008/04/RounderRouter/ ...including an applet, Processing code, Python code, C code, php code, a table of stats, a data file of 2^27 values of f(n)-f(n-1), and an audio version of that data. Jim Propp says of the applet, "one sees (after a few thousand steps) that some of the forms are travelling upward and others are travelling downward, like the angels in Jacob's dream." This is a 1D+time automaton like Wolfram plays with, except it has the property that an incremental change to the initial state causes an incremental change to the whole history. So, there are two nested time dimensions: n, the number of particles that have been added, and what I call t, the row of the triangle. --Steve

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[math-fun] The "square pyramid" puzzle
by Dan Asimov 17 Apr '08

17 Apr '08

This post concerns the uniqueness of the nontrivial solution to (*) 1^2 + 2^2 + 3^3 + . . . + n^2 = K^2 (namely, n = 24 and K = 70), and is largely based on "The square pyramid puzzle" by W.S. Anglin, American Mathematical Monthly, Feb. 1990. The problem seems to have originated with an 1875 challenge from Edouard Lucas to prove that "A square pyramid of cannonballs contains a square number of cannonballs only when it has 24 cannonballs along its base." (He was evidently not counting the trivial solution of n = K = 1.) In 1876 a flawed proof was pubished by M. Moret-Blanc. In 1877 a flawed proof was pubished by Lucas. In 1918 -- 43 years after the problem was posed -- apparently the first valid proof (14 pages) was published, by G.N. Watson, using an extended theory of Jacobi elliptic functions. More proofs appeared in 1952, 1966, and 1975, but not until 1985 did someone (De Gang Ma) publish a proof accessible at the undergraduate level. In a simplification of De Gang Ma's method, in this article Anglin uses 3 lemmas to cover the case of n even, and 7 lemmas (and an entirely different strategy) to cover the case of n odd -- and draws on the theory of Pell's equation and quadratic reciprocity. The arguments involve considering a large number of cases and strike me as quite ad hoc, (although in total they cover only about 3 1/2 pages). My question to math-fun is this: Can this problem really be as difficult as this history would make it seem? Is it possible that considering the equation (*) modulo just the right primes would lead to an efficient proof? Or perhaps there could be a geometric argument? --Dan _____________________________________________________________________ "It don't mean a thing if it ain't got that certain je ne sais quoi." --Peter Schickele

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[math-fun] Hilbert function as images
by Jason 17 Apr '08

17 Apr '08

I wrote up a little program to generate .xpm images of iterations of Hilbert where pixels are of this form: 32 01 Where 0-3 are either shades of gray or R,Y,G,B. Sources, binaries and PNG images here, with both images in their original (mostly tiny) sizes and scaled to a uniform 2048x2048 size: http://lunkwill.org/src/hilbert/ So, for instance, you might go here: http://lunkwill.org/src/hilbert/color/ Then look at these: http://lunkwill.org/src/hilbert/color/scaled-1.xpm.png http://lunkwill.org/src/hilbert/color/scaled-2.xpm.png ...

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Re: [math-fun] a perplexing asymmetry
by James Propp 15 Apr '08

15 Apr '08

Thanks, Steve! >>(1) How big can |f(n) - n/2| be? I suspect it's O(log n). (For >>n up through 500, f(n) - n/2 ranges between 0 and 7/2.) > >It looks sort of log like, but if you graph those numbers above, >there are very piecewise-linear features, which doesn't make me >confident I know what it's doing. Can you post the graph someplace so I can see what you mean? >But the best speedup is to keep the 2D table between calculations. >The change between calculating f(n) and f(n+1) is to increment one >entry on each row. > >So successive visits to a 2D point go down to the right, next time >left, next time right; right, left, right, etc. It's a rounder-router! I remember noticing this, but it didn't occur to me to exploit this for computation. Good idea! >Guess you can do the whole calculation by keeping each >count mod 3. Might be nice to plot in color. Adding a particle to the 2D table and seeing how the colors update might make a fun visual. Jim

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[math-fun] polyomino packing problems
by Erich Friedman 14 Apr '08

14 Apr '08

sorry for tooting my own (website) horn here. for those of you who might be interested in a different sort of polyomino packing problem, there is quite a bit of interest in my math magic problem this month: http://www.stetson.edu/~efriedma/mathmagic/0408.html erich

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[math-fun] "Math ruined my life"
by Dave Dyer 14 Apr '08

14 Apr '08

As seen on TV, a commercial featuring a very down looking long hair, with the voice over "I started doing math...it ruined my life" I guess he really said "Meth".

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[math-fun] "Math ruined my life"
by Dave Dyer 14 Apr '08

14 Apr '08

As seen on TV, a commercial featuring a very down looking long hair, with the voice over "I started doing math...it ruined my life" I guess he really said "Meth".

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Re: [math-fun] crossword puzzle clue in today's (Saturday) New York Times (Byron Walden)
by Dan Asimov 14 Apr '08

14 Apr '08

<< (already yesterday's NYT, I guess, it's after midnight), but I just finished it. Here's the clue: Setting numbered in multiples of the square root of 2 (5 letters) I agonized, but eventually got it by working around it with other clues. >> Same here. (Though in my time zone, Saturday was the day before yesterday.) --Dan _____________________________________________________________________ "It don't mean a thing if it ain't got that certain je ne sais quoi." --Peter Schickele

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