Mathematics Weekly Newsletter - Friday, October 31, 2014

Top new questions this week: Mathematician vs. Computer: A Game A mathematician and a computer are playing a game: First, the mathematician chooses an integer from the range $2,...,1000$. Then, the computer chooses an integer uniformly at random from the same ... (probability) (puzzle) (game-theory)   asked by pew 75 votes answered by martin 42 votes Why do we miss 8 in 0.012345679..., 98 in 0.0001020304050607080910111213..., and so on in fractions like 1/81, 1/9801, and so on? I've seen this happen that when you divide by a fraction using the square of any set of nines in the denominator depending on how many there are like ${1\over 99^2}={1\over 9,801}$, you get ... (fractions)   asked by Mathster 41 votes answered by mapierce271 50 votes Geometric mean of reals between 0 and 1 What is the geometric mean of all reals between $0$ and $1$? I was thinking over this, but could not come up with anything useful. Please help me out. (means)   asked by Pkwssis 25 votes answered by David 40 votes Prove the integral evaluates to $\frac{K}{\pi}$ Yesterday I received the following integral that might require some tedious steps to do $$\int_0^{\infty}{\small\left[ \frac{x}{\log^2\left(e^{\large x^2}-1\right)}- \frac{x}{\sqrt{e^{\large ... (calculus) (integration) (definite-integrals) (improper-integrals)   asked by Chris's sis 20 votes answered by Winther 3 votes Given any computable number, is there any algorithm to decide whether it is transcendental? Given any computable number $a_c$, is there any algorithm to decide whether it is transcendental? Definition of "computable number": According to Ming Li and Vitanyi, a real number $x=0.x_1x_2\ldots$ ... (number-theory) (computability)   asked by XL _at_China 18 votes answered by TonyK 29 votes simple quadratic, crazy question Have been working on this for years. Need a system which proves that there exists a number $C$ which has certain properties. I will give a specific example, but am looking for a system which could ... (number-theory)   asked by Aaron Horak 16 votes answered by J. David Taylor 11 votes speaking about math How should one vocalise difficult terms, which have no short names, in a talk? Have you other hints for talking about maths to other people? Do I need all to mention, what I write down on the ... (soft-question)   asked by bjn 15 votes answered by Jair Taylor 21 votes Greatest hits from previous weeks: Pedagogy: How to cure students of the "law of universal linearity"? One of the commonest mistakes made by students, appearing at every level of maths education up to about early undergraduate, is the so-called "Law of Universal Linearity": $$ \frac{1}{a+b} ... (algebra-precalculus) (education) (teaching)   asked by Peter LeFanu Lumsdaine 225 votes answered by goblin 113 votes Teenager solves Newton dynamics problem - where is the paper? From Ottawa Citizen (and all over, really): An Indian-born teenager has won a research award for solving a mathematical problem first posed by Sir Isaac Newton more than 300 years ago that has ... (differential-equations) (physics) (classical-mechanics)   asked by jnm2 54 votes answered by Américo Tavares 30 votes Can you answer these? Books to read to understand Terence Tao's Analytic Number Theory Papers I tried to understand Terence Tao's Analytic Number Theory Papers. For example, this paper, Every Odd Number Greater Than 1 is The Sum of at Most Five Primes. Which books shall I read to prepare ... (number-theory) (elementary-number-theory) (reference-request) (analytic-number-theory) (goldbachs-conjecture)   asked by jim 6 votes Finding the closed form of $\sum_{k=1}^{\infty} \sum_{n=1}^{\infty}(-1)^{k+n} \frac{\log(k+n)}{k n}$ A while ago I computed pretty easily the series $\sum_{k=1}^{\infty} \sum_{n=1}^{\infty}(-1)^{k+n} \frac{\log(k+n)}{k+ n}$ and then I thought of tackling the case where we have the product instead of ... (calculus) (real-analysis) (integration) (sequences-and-series) (improper-integrals)   asked by Chris's sis 5 votes Open mathematical questions for which we really, really have no idea what the answer is There is no shortage of open problems in mathematics. While a formal proof for any of them remains elusive, with the "yes/no" questions among them mathematicians are typically not working in both ... (soft-question) (big-list) (conjectures) (open-problem)   asked by pew 12 votes

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