September 2006 - mathfun@mailman.xmission.com

Re: Solution Re: [math-fun] 1,2,3,5,18 puzzle
by Daniel Asimov 14 Sep '06

14 Sep '06

Fred wries: << A few years ago a local schoolteacher called Neil Hamilton observed that the original Collatz problem, for the function f(x) = 3x+1 if x is odd, x/2 if x is even is equivalent to the analogous problem for the new function g(y) = (y+1)/4 if y mod 4 = 3, (3y+1)/4 if y mod 4 = 1, 3y/2 if y mod 4 = 0,2. While I said at the time that I thought this was probably well-known, I didn't find any mention of it in the literature; and it did occur to me to wonder whether there had been any systematic attempt to investigate such equivalences. >> Yes, this is well-known, and sometimes the way that Jeff Lagarias prefers to define the Collatz function. (My own preference is to let f(N) = (3N+1)/(the largest power of 2 that divides 3N+1). which means the domain and range may be considered to be the odd integers.) --Dan

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Re: Solution Re: [math-fun] 1,2,3,5,18 puzzle
by Marc LeBrun 14 Sep '06

14 Sep '06

Heh! Actually, I've found a marvelous cycle of length 442783582734897230450239994058234089230450023451234951 but the margin is too small to contain the start value! More seriously, it's always seemed to me that Collatz might be a member of some parametric family--including others like 3n-1, 3n+5, 17n+7, 105n+69...say. Then a "real" answer would be a theory that predicts the cycle structure given the parameters--as opposed to just this one instance. Or, perhaps at least a negative result such as that it's incomputably chaotic...

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Correction Re: [math-fun] algebra question
by Daniel Asimov 13 Sep '06

13 Sep '06

Which should have read: < Bill Cordwell writes: << According to Grove, there are nine rings of algebraic integers over Q[sqrt(m)] that are PIDs, where m is a negative integer, viz., m = -1,-2,-3,-7,-11,-19,-43,-67, and -163. These are then UFDs, but are they the only such UFDs? >> Yes, the only UFD's of the form Z[sqrt(-n)] for a positive integer n. -- (NOT "the only PID's . . . ") I believe that, amazingly, Gauss conjectured this (or at least had found all 9 and no others). It was proven by Heegner in the early 1950s. His proof was erroneously believed to have flaws, and it was re-proved by Baker & Stark (around 1970?). At some later point it was recognized that Heegner's proof was correct after all. >

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Re: Solution Re: [math-fun] 1,2,3,5,18 puzzle
by Daniel Asimov 13 Sep '06

13 Sep '06

Marc writes: << Heh! Actually, I've found a marvelous cycle of length 442783582734897230450239994058234089230450023451234951 but the margin is too small to contain the start value! More seriously, it's always seemed to me that Collatz might be a member of some parametric family--including others like 3n-1, 3n+5, 17n+7, 105n+69...say. Then a "real" answer would be a theory that predicts the cycle structure given the parameters--as opposed to just this one instance. Or, perhaps at least a negative result such as that it's incomputably chaotic... >> In fact, Jeff Lagarias has painstakingly annotated every paper on the subject he could get his hands on (and at least listed the others!) in two bibliographies, one up through 2000, and another 2001-2006, which are easily googlable with the string lagarias collatz and freely downloadable in several formats. A number of suggestions related to Marc's have been given some attention. (Note: replacing 3n+1 by 3n-1 is equivalent to interchanging the roles of positive and negative integers.) --Dan

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Solution Re: [math-fun] 1,2,3,5,18 puzzle
by Daniel Asimov 13 Sep '06

13 Sep '06

<< << Puzzle: What do the positive integers 1, 2, 3, 5, 18 have in common with each other, but with no other positive integer (conjecturally) ? >> Hint: These number are cycle lengths. >> ---------------------------------------------------------------------------- The answer is: 1, 2, 3, 5, 18 are the cycle lengths of the 5 known (and conjecturally, only) cycles of the Collatz function f: Z -> Z when iterated, where f(n) is defined as f(n) = n/2 if n is even; f(n) = 3n+1 if n is odd. The cycles -- of lengths 1, 2, 3, 5, 18 -- are those containing 0, -1, 1, -5, -17, respectively. --Dan

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Re: [math-fun] algebra question
by Daniel Asimov 13 Sep '06

13 Sep '06

Bill Cordwell writes: << According to Grove, there are nine rings of algebraic integers over Q[sqrt(m)] that are PIDs, where m is a negative integer, viz., m = -1,-2,-3,-7,-11,-19,-43,-67, and -163. These are then UFDs, but are they the only such UFDs? >> Yes, the only PID's of the form Z[sqrt(-n)] for a positive integer n. I believe that, amazingly, Gauss conjectured this (or at least had found all 9 and no others). It was proven by Heegner in the early 1950s. His proof was erroneously believed to have flaws, and it was re-proved by Baker & Stark (around 1970?). At some later point it was recognized that Heegner's proof was correct after all. --Dan

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[math-fun] article by barry mazur on sato tate conjecture
by Thane Plambeck 13 Sep '06

13 Sep '06

It's in this week's Nature magazine (normally subscription only) but there's a copy of it on his website http://www.math.harvard.edu/~mazur/papers/nature_sato_tate.pdf -- Thane Plambeck http://www.plambeck.org/ehome.htm

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Re: [math-fun] mis-defining limits
by Daniel Asimov 12 Sep '06

12 Sep '06

Richard Guy asks: << Is there any difficulty as x --> 0- or 0+ ? I thought that the interesting question is x --> 1- R. . . . On 9/10/06, Daniel Asimov <dasimov(a)earthlink.net> wrote: < Btw, on his website, Noam Elikies asks what the limit is of the infinite series g(x) = x - x^2 + x^4 - x^8 + . . . +- x^(2^k) -+ . . . (which converges for |x| < 1) as x -> 0-. It's a cool problem to let students try to guess the answer just using calculators. > >> Good point, Richard, and the limit as x -> 0 is not that interesting, either. I posted the wrong question, yet another mistake created by undue haste. *****The question of interest is: "What happens to g(x) as x -> 1- ?" --Dan

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Re: [math-fun] mis-defining limits
by Daniel Asimov 11 Sep '06

11 Sep '06

<< I'm teaching an honors calculus class this semester, and to motivate the formal epsilon-delta definition of limits, I'd like to show the students some appealing, informal definitions of limits and the hidden errors and ambiguities of those definitions. Do any of you have any favorite examples of this? >> If you can first use (informal) limits to show that the slope of the tangent line T_x to y = x^2 at (x,x^2) is 2x, then an interesting limit is the limit of the y-intercept of T_c as c -> 0. A similar example is the limit of the area of the triangle formed by the normal line N_c to y = x^3 at (c, c^3), and the x- and y-axes, as c -> 0+. << E.g.,, I seem to recall that some definitions of limits say that f(x) should approach but never equal b as x approaches a (which would make lim_{x -> 0} x sin x problematical). >> To say nothing of limits of constant functions! I've never heard this condition, but such a notion is SO WRONG that I wouldn't even put it into student's heads. *Especially* since there is a very slightly related condition that IS part of the definition of a limit: Given f: R -> R, say, the (limit of f(x) as x -> a) = L means (and of course you know the following; it's just the way I like to define a limit when we finally get around to the formal def. -- in four easily understood parts: ---------------------------------------------------------------------------------------- For all e > 0 there exists a number d > 0 (that may depend on e) such that if 0 < |x - a| < e then |f(x) - L| < e. ------------------------------------------------------------------------------------------ It is the lower left condition, in particular the 0 < |x - a| part of the inequality, that a student could easily confuse with the very wrong condition "but never equal b" that you mention, Jim. I hope you never once bring it up. Btw, on his website, Noam Elikies asks what the limit is of the infinite series g(x) = x - x^2 + x^4 - x^8 + . . . +- x^(2^k) -+ . . . (which converges for |x| < 1) as x -> 0-. It's a cool problem to let students try to guess the answer just using calculators. --Dan

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[math-fun] (q;q)_oo (Dedekind eta) approximation
by R. William Gosper 11 Sep '06

11 Sep '06

In http://arxiv.org/abs/math/0212035, Alan Sokal (of Social Text infamy) uses (q;q)_oo ~ e^(pi^2/ln q). Far closer (1% for -.23<q<1): li (sqrt(q)) 2 ------------ log(q) %e (q; q) = ----------------- inf 1 ---------------- 1 24 (------- - 1) sqrt(q) q Note the strange whiff of q^(1/24) even though it wasn't in the product. Mma draws a nice lily pad for ParametricPlot3D[{r*Cos[\[Theta]], r*Sin[\[Theta]], Abs[DedekindEta[Log[q]/2/I/Pi]/q^(1/24) - E^(PolyLog[2, Sqrt[q]]/Log[q] + Log[q]/24/(1 - 1/Sqrt[q])) /. q -> r*E^(I*\[Theta])]}, {r, .01, .99}, {\[Theta], 0, 2Pi}] (Mma defines DedekindEta[tau], not [q].) --rwg GASOLINIC LOGICIANS

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