August 2004 - mathfun@mailman.xmission.com

[math-fun] The top t tennis team.
by David Gale 09 Sep '04

09 Sep '04

Given n candidates trying out for a tennis team of t players. Find the minimum number of matches needed to pick out the top t. I don't expect anyone to solve this. The related problem of picking the t'th best player remains unsolved for n >7. (see Knuth, vol. 3)What I noticed is, 1). The top t problem is strictly "easier" than the t'th best problem. Example. You can find the top 2 out of 5 in only 5 matches, (try it), whereas finding the second best requires 6 matches. 2).There doesn't seem to be any treatment of the top t problem in the literature. Knuth doesn't even mention it. References, anyone?

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[math-fun] No politics on Math-Fun, please
by Richard Schroeppel 30 Aug '04

30 Aug '04

Newcomers to Math-Fun may be unaware that political messages to Math-Fun are discouraged. This is to keep the peace. Thane has been placed on "message moderation" for 30 days. Rich rcs(a)cs.arizona.edu PS: If you feel compelled to comment on this, please send it to me, not the group. Most subscribers are here for the math.

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[math-fun] An Amazing Anagram
by victor＠idaccr.org 30 Aug '04

30 Aug '04

From one of those circulating Internet lists: "To be or not to be: that is the question, whether it's nobler in the mind to suffer the slings and arrows of outrageous fortune." ANAGRAM: "In one of the Bard's best-thought-of tragedies, our insistent hero, Hamlet, queries on two fronts about how life turns rotten." Victor P.S. I check it, just to make sure.

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[math-fun] lewis carroll scrapbook digitized at the library of congress
by Thane Plambeck 30 Aug '04

30 Aug '04

Not much math in this, but some insight into the alice character http://international.loc.gov/intldl/carrollhtml/ -- Thane Plambeck thane(a)best.com http://www.plambeck.org

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[math-fun] q-analogue of MacMahon's plane partition formula
by wouter meeussen 28 Aug '04

28 Aug '04

.. as mentioned at the end of Diagonal Sums of Boxed Plane Partitions www.combinatorics.org/Volume_8/PDF/v8i1n1.pdf refering to M.D. Blum , unpublished notes (1996) could anyone explain how to interpret the 'sub_q' in (i+j+k-1)_sub_q. I would like to express the number of plane partitions ( of n ) that *just* fit inside a k by k by k box, not the total count irrespective of n as in McMahon's formula. Wouter.

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[math-fun] Re: Infinitely many primes, maybe
by Schroeppel, Richard 25 Aug '04

25 Aug '04

I wrote > I don't know of any adjustments for N!+1 except divisor>N, > but would be happy to learn. Ed Pegg reminded me of Wilson's Theorem, that P | (P-1)!+1 iff P is prime. Knowing that a number has no small divisors gives a modest boost to its prime-likelihood. There's enough data on N!+1 primes to compare reality with prediction. Rich

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RE: [math-fun] Infinitely many primes, maybe
by Schroeppel, Richard 23 Aug '04

23 Aug '04

There are a couple of "fine points" to be mentioned here. There are odd numbers K such that 2^N+K is composite for all large N. I think the smallest such K is about 75000. The proof is elementary: Split N = 24 M + R, 0<=R<24, and check that each R leads to a small factor. [I may be wrong about 24, it could be 12 or 16.] 2^N+1 is special because N cannot have odd divisors > 1, and must itself be a power of 2. 2^N+3 and its siblings will need a correction factor relative to typical numbers of the same size: Never divisible by 2 or 3, 1/4 chance of 5-factor, 2/6 of 7-factor, 1/10 of 11-factor, 1/12 of 13-f, never 17, etc. We assume that the product of the correction factors is convergent, making only a modest change to the expected number of primes. 2^N+9 is either the sum of two squares or A^2+2B^2; this excludes divisors 8K+7. 2^N-9 must have N odd to be prime, excluding divisors 8K+-3. 2^P-1 has additional complications: No divisor < 2P+1; divisors must be 2kP+1 and also 8K+-1. On the other hand, divisors of these forms seem to have matching increases in likelihood. I don't know of any adjustments for N!+1 except divisor>N, but would be happy to learn. Rich -----Original Message----- From: David Gale To: math-fun(a)mailman.xmission.com Sent: 8/10/2004 5:08 AM Subject: [math-fun] Infinitely many primes, maybe In the excellent article on Mersenne primes http://www.utm.edu/research/primes/mersenne/ <http://www.utm.edu/research/primes/mersenne/> I found the following, to me, cryptic passage "Are there infinitely many Mersenne primes? The answer is probably[ital] yes (because the harmonic series diverges). Could someone please explain the "because"? Does the same argument work for Fermat primes? for primes of the form 10^n + 3? for (a^ n)+b, gcd[a,b]=1?

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[math-fun] number derivatives
by Mike Stay 20 Aug '04

20 Aug '04

I had some fun with number derivatives and wrote a paper on it: http://www.cs.auckland.ac.nz/~msta039/numderiv.ps I'd appreciate comments and, if you think it's worthy of publication, names of appropriate journals. Thanks! -- Mike Stay staym(a)clear.net.nz http://www.cs.auckland.ac.nz/~msta039

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[math-fun] ignorance
by wouter meeussen 19 Aug '04

19 Aug '04

does (-1+2^Prime[n]) ever divide (Prime[2^n]!) (apart from n=1 and 2 ?) W.

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[math-fun] about Eulerian numbers 2^n-n-1
by wouter meeussen 19 Aug '04

19 Aug '04

hi all, is it evident that A000295 in EIS, defined as in title, also equals the exponent of 2 in (2^n-1)! as calculated by Table[ pow[ 2^n - 1 , 2], {n, 29} ] using pow[ nfac_ , p_] := (nfac - Plus @@ IntegerDigits[ nfac, p ])/(p - 1) not earth-shaking, just in need of 'sanity check' Wouter.

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