November 2002 - mathfun@mailman.xmission.com

Correction Re: [math-fun] Sums of four squares
by asimovd＠aol.com 21 Nov '02

21 Nov '02

I thank Dan Hoey for pointing out a basic mistake in my statement << Define the n-onion O_n as the union of all spheres S_k(n) about 0 in R^n whose radius is sqrt(k) for any integer k >= 0 (including the degenerate sphere S_0(n)). >> The last statement should read: Then because of the sharpness of the 4-square theorem, for each n = 1,2,3, some spheres S_k(n) will not intersect Z^n, while for n >= 4,every S_k(n) will intersect Z^n. Dan A.

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Re: [math-fun] Sums of four squares
by asimovd＠aol.com 21 Nov '02

21 Nov '02

Define the n-onion O_n as the union of all spheres S_k(n) about 0 in R^n whose radius is sqrt(k) for any integer k >= 0 (including the degenerate sphere S_0(n)). Then because of the sharpness of the 4-square theorem, O_1, O_2, and O_3 will not contain all the integer points in their respective spaces, while O_n for n >= 4 will. QUESTION: As long as we're discussing enumerating representations of 3-square integers, what about the 4-square ones (i.e., all integers) ? For a given k, how many points P(k) of the integer lattice in R^4 will have radius sqrt(k), i.e., lie on S_k(4) ??? --Dan

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[math-fun] 20/11/02
by Ken Roberts 21 Nov '02

21 Nov '02

Hi Everyone, Just realized that, European style, today 20/11/02 is a neat date. Cheers, Ken R.

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Re: [math-fun] 20/11/02
by Dan Hoey 21 Nov '02

21 Nov '02

Richard Guy wrote: > So is 02-11-20, and more logical, and consistent > with an agreement signed by almost every country > in the world a little over a quarter of a century > ago, even including the US at that time.... Indeed, it is self-consistent in the sense advocated by Danhy Cohen in IEN137, "On Holy Wars and a Plea for Peace". While I notice that the note is dated 1 April 1980 [sic], I believe the case he makes for endian consistency is sincere. Also, I note that 02-11-20 refers to the same date whether we use the standard form (year=02, month=11, day=20) or the self-consistent little-endian notation (day=02[le], month=11[le], year=20[le]). Of course, the year is more properly written "2002" (or "2002[le]") so the really neat dates start almost eight years from now. Dan Hoey Hoey(a)AIC.NRL.Navy.Mil

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[math-fun] Sums of three squares
by Eugene Salamin 21 Nov '02

21 Nov '02

There is a nice way to calculate the number r2(n), n>0, of integer solutions (x,y) of x^2 + y^2 = n. Factor n as n = p1^a1 p2^a2 ... q1^b1 q2^b2 ... 2^c, where the p's are primes == 1 mod 4, and the q's are primes == 3 mod 4. Then r2(n) = if any b is odd, then 0 else, 4 (1 + a1) (1 + a2) ... . Is there a similar formula for the number r3(n) of integer solutions (x,y,z) of x^2 + y^2 + z^2 = n ? __________________________________________________ Do you Yahoo!? Yahoo! Web Hosting - Let the expert host your site http://webhosting.yahoo.com

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Re: [math-fun] Steven Krantz's review of A New Kind of Science
by Dan Hoey 20 Nov '02

20 Nov '02

I'm amused by Krantz's report that Wolfram tells us that he has entered one hundred million keystrokes on his computer in the past ten years by way of creating the scientific work that we now read[1]. and the footnote [1] If you send twenty brief e-mail messages per day, you will find that you enter over thirty-five million keystrokes in ten years. If you do things like write books or articles, edit journals, conduct professional correspondence, write reviews, you may find that you actually break Wolfram's keystroke record. Since computer programming and research are also keyboard- intensive activities, I can't help musing on how many keystrokes were entered by the researchers whose results Wolfram has reported. Dan Hoey Hoey(a)AIC.NRL.Navy.Mil

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Re: [math-fun] Origins of a problem.
by POPPY9X＠aol.com 20 Nov '02

20 Nov '02

there is an old problem in which one asks that the integers from 1 to N be listed so that the sum of adjacent integers are primes. I've never seen a solution. It is probably true that one can always do this with N at the end of the list. if one knew that for N>9 there is always a pair of twin primes betwen N and 2N all of this would follow. But there should be a real proof. anyone know anything?

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Re: [math-fun] Origins of a problem.
by POPPY9X＠aol.com 20 Nov '02

20 Nov '02

the original problem was not "circular" but just a list for example with N=12 we have: 12, 7, 10. 9. 8. 11, 6, 5, 2, 1, 4, 3, down thru 6 illustrates the twin prime connection (here the primes are 17 and 19)_ it also provides an inductive procedure for verifying the conjecture for N up to the linit of tables of primes (millions)

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RE: [math-fun] funny little limit on central binomial coefficient s
by Meeussen Wouter (bkarnd) 20 Nov '02

20 Nov '02

thanks Shel, Rich's shortish answer helped me grok it in half a day (he must have seen it in half a second): ratio of successive central binomial coefficients = 4-2/(1+n) (running sum of seq ...), seq[n], seq[n+1] x + 1 = 4 (solve for x) (* same goes for catalans *) cute no? that's why I like this group. Wouter Meeussen email: wouter.meeussen(a)vandemoortele.com =============================== This email is confidential and intended solely for the use of the individual to whom it is addressed. If you are not the intended recipient, be advised that you have received this email in error and that any use, dissemination, forwarding, printing, or copying of this email is strictly prohibited. You are explicitly requested to notify the sender of this email that the intended recipient was not reached.

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[math-fun] five, six, and eight (was: seven)
by James Propp 20 Nov '02

20 Nov '02

Last week, I wrote (in response to Dan Hoey's posting): >My mistake; as Dan Asimov pointed out, the sequence has period 5. (thereby showing that I can confuse one Dan with another with as much facility as I can confuse small natural numbers :-) ). I should mention that the theory of cluster algebras currently being developed by Fomin and Zelevinsky gives a context for these facts, but it doesn't really explain them in an elementary way. Here is a restatement of the "order-5 fact", along with two other facts of a similar kind: 1) Any sequence x(1),x(2),x(3),... defined by the recurrence x(n) = (x(n-1)+1)/x(n-2) (for n>2) has period 5. 2) Any sequence x(1),x(2),x(3),... defined by the recurrence { (x(n-1)+1)/x(n-2) (for n>2, n odd) x(n) = { { (x(n-1)^2+1)/x(n-2) (for n>2, n even) has period 6. 3) Any sequence x(1),x(2),x(3),... defined by the recurrence { (x(n-1)+1)/x(n-2) (for n>2, n odd) x(n) = { { (x(n-1)^3+1)/x(n-2) (for n>2, n even) has period 8. (Note that this is different from the period-8 recurrence I mentioned in my previous email.) In the theory of cluster algebras, these recurrences correspond to the Lie algebras A_2; B_2 (or C_2); and G_2 (respectively). But I'm just quoting someone else here; I don't actually understand the link between Lie algebras and cluster algebras. Nor do I have the foggiest notion of whether the x(n) = (x(n-1)+x(n-2)+1)/x(n-3) recurrence fits into the story somehow. But since there's such a nice story for the fact that the map t -> 1-1/t has order 3, it doesn't seem far-fetched to hope for something similar in these cases. Jim Propp

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