December 2015 - mathfun@mailman.xmission.com

[math-fun] exp(Pi)- Pi and then ?
by Simon Plouffe 05 Dec '15

05 Dec '15

Hello funsters, as you may know, there is one funny approximation of 20 with exp(Pi) - Pi = 19.999099979... but are there others the same ? maybe, by using high speed guessing and some tables, 3 - (5*Pi)/6 - 6/(5*Pi) = 0 (the error is 1/29189). 2*Pi/5 + exp(-2*Pi/5) - 5/3*Pi -log(4*Pi/5) - i*Pi = 22/7 (absolute value) with 12 digits of precision. (use fixed fonts) : Pi 8 Pi Pi ---- + exp(- ----) + ln(----) + exp(Pi) + Pi I 3 5 2 = 24 + 93/110 , 11 digits precision. that is : 1/3*Pi+exp(-8/5*Pi)+ln(1/2*Pi)+exp(Pi)+I*Pi, absolute value. any patterns ? aby clue ? have a nice day, Simon Plouffe

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[math-fun] phi(N)*sigma(N)<N^2, phi(N)+sigma(N)>=2*N
by Warren D Smith 02 Dec '15

02 Dec '15

The formula for phi(N) is N * product (1 - 1/P) product taken over primes P that divide N. The formula for sigma(N) is N * product (1 + 1/P + 1/P^2 + ... 1/P^E) with P dividing N and P^E being the highest power of P that divides N. So obviously sigma(N) < N * product(1 + 1/P + 1/P^2 + ... ) = N^2 / phi(N). Q.E.D. Also obviously sigma(N) >= N * product ( 1 + 1/P ) so phi(N) + sigma(N) >= 2*N since the cross terms in the products all either cancel out or come out to something positive. [The avoidance of phi+sigma=2*N+5 is not so trivial, though.] -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)

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[math-fun] How random are (real) dice?
by Henry Baker 01 Dec '15

01 Dec '15

FYI -- Humans have been playing with these things for 3400+ years; now we finally get around to measuring them? https://en.wikipedia.org/wiki/Dice http://www.markfickett.com/stuff/artPage.php?id=389 An *automatic system* for rolling a polyhedral die and taking photos of the rolls; extracting the image of just the die from those images; clustering the images of the die by which face is shown; and analyzing the results.

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[math-fun] Fwd: McIntosh's death
by Fred Lunnon 01 Dec '15

01 Dec '15

Via Genaro Martinez --- << Dear friends and colleagues, I hope everybody are very fine, it is just for you possible interest. Today, Prof. Harold V. McIntosh dead this night in Puebla, Mexico to 86 years old. Harold V. McIntosh 11.05.1929 Denver, Colorado, USA. 30.11.2015 Puebla, Puebla, Mexico. Warmest regards, Genaro >>

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