March 2010 - mathfun@mailman.xmission.com

[math-fun] A prime factor of the Fermat number F22
by Warut Roonguthai 27 Mar '10

27 Mar '10

Someone has just found a 35-digit prime factor of F22. This is less than two months after a factor of F14 was discovered. The cofactor is composite. See http://mersenneforum.org/showthread.php?p=209627 Now F20 and F24 are the only known composite Fermat numbers without any known prime factors. Warut

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[math-fun] q-binomial-oids (ascii art included)
by Bill Gosper 26 Mar '10

26 Mar '10

I can find the following in the literature: > ___n-1 M-k > | | (1-q ) > ___M-1 n \~~ M | |k=0 n n (n-1)/2 > | | (1+x q ) = > -------------- x q > | |n=0 /__ n=0 ___n k > | | (1-q ) > | |k=1 Now I came up with > ___n-1 M-k > | | (1-q ) > 1 \~~ M | |k=0 n n (n-1) > -------------- = > ------------------------------- x q > ___M-1 n /__ n=0 ___n-1 k ___n-1 k > | | (1-x q ) | | (1-q q ) | | (1-x q ) > | |n=0 | |k=0 | |k=0 This is certainly known. Can anyone point out where this is given? Herr Gosper? LaTeX sources are \prod_{n=0}^{M-1}{(1+x\,q^n)} & = & \sum_{n=0}^{M}{ \frac{\prod_{k=0}^{n-1}{(1-q^{M-k})}} {\prod_{k=1}^{n}{(1-q^k)}} \, x^n \, q^{n\,(n-1)/2} } and \frac{1}{\prod_{n=0}^{M-1}{(1-x\,q^n)}} & = & \sum_{n=0}^{M}{ \frac{\prod_{k=0}^{n-1}{(1-q^{M-k})}} {\prod_{k=0}^{n-1}{(1-q\,q^k)} \, \prod_{k=0}^{n-1}{(1-x\,q^k)}} \, x^n \, q^{n\,(n-1)} } Modulo some changes of variable (e.g. to obviate x), and some product massaging, that's the q-Chu-Vandermonde sum, Basic Hypergeometric Series, Appendix II, II.7 . This came up here 8 nov 05, when I derived the known result: >the probability that an nxn bitmatrix will have (mod 2) rank = k is 2 n - k + 1 k + 1 (n - k) (q ; q) (q ; q) q k n - k >P(n,k) := -------------------------------------------, (q; q) n - k > if the entries are 1 with probability p = 1-q . More generally, for an m x n matrix, m - k + 1 k + 1 (m - k) (n - k) (q ; q) (q ; q) q k n - k P(m, n, k) = --------------------------------------------------. (q; q) n - k --rwg

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[math-fun] Circled and rounded perspective
by Olivier Gerard 20 Mar '10

20 Mar '10

Reminiscent of plane models of non euclidian geometry In this recent post of 0xDE http://11011110.livejournal.com/193877.html I like the effect of perspective 1 and 4. Any comments and further references ? Olivier

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[math-fun] Millennium Prize Awarded For Perelman's Poincaré Proof
by Ray Tayek 19 Mar '10

19 Mar '10

http://science.slashdot.org/story/10/03/19/0540249/Millennium-Prize-Awarded… "The Clay Mathematics Institute has <http://www.claymath.org/poincare/>announced its acceptance of Dr. Grigori Perelman's proof of the Poincaré conjecture and awarded the first Millennium Prize. Poincaré questioned whether there exists a method for determining whether a three-dimensional manifold is a spherical: is there a 3-manifold not homologous to the 3-sphere in which any loop can be gradually shrunk to a single point? The Poincaré conjecture is that there is no such 3-manifold, i.e. any boundless 3-manifold in which the condition holds is homeomorphic to the 3-sphere. A <http://en.wikipedia.org/wiki/Solution_of_the_Poincar%C3%A9_conjecture>sketch of the proof using language intended for the lay reader is available at Wikipedia." ... --- co-chair http://ocjug.org/

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[math-fun] q-binomial-oids (ascii art included)
by Joerg Arndt 19 Mar '10

19 Mar '10

I can find the following in the literature: > ___n-1 M-k > | | (1-q ) > ___M-1 n \~~ M | |k=0 n n (n-1)/2 > | | (1+x q ) = > -------------- x q > | |n=0 /__ n=0 ___n k > | | (1-q ) > | |k=1 Now I came up with > ___n-1 M-k > | | (1-q ) > 1 \~~ M | |k=0 n n (n-1) > -------------- = > ------------------------------- x q > ___M-1 n /__ n=0 ___n-1 k ___n-1 k > | | (1-x q ) | | (1-q q ) | | (1-x q ) > | |n=0 | |k=0 | |k=0 This is certainly known. Can anyone point out where this is given? Herr Gosper? LaTeX sources are \prod_{n=0}^{M-1}{(1+x\,q^n)} & = & \sum_{n=0}^{M}{ \frac{\prod_{k=0}^{n-1}{(1-q^{M-k})}} {\prod_{k=1}^{n}{(1-q^k)}} \, x^n \, q^{n\,(n-1)/2} } and \frac{1}{\prod_{n=0}^{M-1}{(1-x\,q^n)}} & = & \sum_{n=0}^{M}{ \frac{\prod_{k=0}^{n-1}{(1-q^{M-k})}} {\prod_{k=0}^{n-1}{(1-q\,q^k)} \, \prod_{k=0}^{n-1}{(1-x\,q^k)}} \, x^n \, q^{n\,(n-1)} }

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[math-fun] Earliest Lisp program ?
by Henry Baker 18 Mar '10

18 Mar '10

The discussion of the 40-year anniversary of the Internet in Oct'09 talked about the first message ever sent: "lo" (the first two characters of "login"; the machine crashed before "g"!). I was wondering what was the first non-trivial (i.e., with a loop or recursion) Lisp function ever executed. I would be willing to guess "APPEND", because I think that the first Lisp didn't have numbers, so "FACT" (factorial) wasn't in the running. Does anyone here know the answer?

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[math-fun] MIT admissions as easy as pi
by Henry Baker 17 Mar '10

17 Mar '10

FYI -- http://www.boston.com/news/local/breaking_news/2010/03/mit_admissions.html The Quad MIT admissions as easy as pi March 17, 2010 04:50 PM By Tracy Jan, Globe Staff Last Sunday afternoon, at precisely 1:59, every MIT applicant worth their salt logged on to a special admissions website to check their fates. Just 1,611 students got in, a record low admit rate of 9.7 percent. But a weightier number filled studentsÂ minds: 3.14159Â, or pi, the Greek letter symbolizing the ratio of the circumference of a circle to its diameter. Yes, in true Geek fashion, MIT released its admissions decisions at a time that would hold a deeper meaning for only those mathematically inclined enough to apply Â on Pi Day, March 14, 1:59, as an homage to the mathematical constant. "We thought it was pretty neat and fitting,'' said Stuart Schmill, dean of admissions and a 1986 MIT graduate. "Pi is a special number. And this is one way for us, as a science and technology centered place, to celebrate it." Call it MITÂs version of March Madness. Applicants were e-mailed a clever note last Thursday, instructing them how to check the status of their applications Âa minute before 2 p.m.Â ÂReceiving your decision online is as easy as pi,Â the message said. Though there was no explicit reference to Pi Day, Schmill said, "Most of our applicants are clued in." This is the second year in a row that the university has been able to release its decisions on Pi Day (coincidentally the birthday of Albert Einstein). But MIT is not making Pi Day admissions -- and rejections -- a tradition just yet. It lands on a Monday next year. And, Schmill said, "We wonÂt release decisions when our applicants are in school."

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[math-fun] What is a "k-vector"?
by Fred lunnon 16 Mar '10

16 Mar '10

On the GA list http://groups.google.com/group/geometric_algebra/ I recently claimed that the term "k-vector" (as in 2-vector, 3-vector) routinely denotes a vector with dimension k. This was subsequently queried, whereupon I was obliged to admit my inability to quote any actual example from the literature. Can anybody clarify the situation --- or better still, provide some relevant citations? Fred Lunnon

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[math-fun] Class number query
by David Wilson 16 Mar '10

16 Mar '10

Let f(n) = SUM(0 <= k < n; (k^2) mod n) where mod is the remainder function. It looks as if, for prime p >= 5 f(p) = 1 if p == 1 (mod 4) 2c+1 if p == 3 (mod 4) where c is the class number of Q(sqrt(-p)) Is this true? Well known?

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[math-fun] oddity
by Bill Gosper 15 Mar '10

15 Mar '10

Shew that odd 2 i k pi 2 i k pi ==== -------- -------- \ odd odd > e oddfun(e ) / ==== k = 1 odd i (2 k - 1) pi i (2 k - 1) pi ==== -------------- -------------- \ odd odd = > e oddfun(e ) / ==== k = 1 Easy but not trivial. --rwg

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