User:GFauxPas/Sandbox

Welcome to my sandbox, you are free to play here as long as you don't track sand onto the main wiki. --GFauxPas 09:28, 7 November 2011 (CST)

User:GFauxPas/Sandbox/Zeta2/lnxln1-x/existence

User:GFauxPas/Sandbox/Zeta2/lnxln1-x/integrand

User:GFauxPas/Sandbox/Zeta2/lnxln1-x/evaluation

User:GFauxPas/Sandbox/Zeta2/FourierSeries/

User:GFauxPas/Sandbox/Zeta2/Informal Proof

Problem Set
All hints are welcome.


 * $\checkmark (1):$ Prove Greatest Power of Two not Divisor. Use this fact to prove Harmonic Numbers not Integers.


 * $(2):$ Prove that:


 * $\displaystyle K_n = 1 + \frac 1 3 + \frac 1 5 + \ldots + \frac 1 {2n+1}$

is not an integer for $n > 1$.

Hint: The proof will be similar to the solution to $(1)$:

Consider the set $\omega = \left \{ {1, 3, \ldots, 2n+1}\right\}$ and let $3^d$ be the highest power of $3$ in $\omega$. Prove that $3^d$ is not a divisor of any other integer in $\omega$ and use this fact to prove $K_n$ is not an integer.


 * $(3):$ Prove that $\dfrac {\ln 2}{\ln 3}$ is irrational. For integers $p$ and $q$, what condition is essential for $\dfrac {\ln p}{\ln q}$ to be irrational? Justify your claims.

(I'm expected to use external sources for this one.)


 * $\dfrac {\ln 2}{\ln 3} = \log_3 2$ from an elementary result you'll find on somewhere (Change of Base of Logarithms or something).  Then you need Irrationality of Logarithm which Ybab321 posted up a couple of months ago. --prime mover (talk) 21:06, 23 November 2014 (UTC)


 * Well that's a lot more direct than I expected it to be, thanks a lot, you and Ybab. --GFauxPas (talk) 21:14, 23 November 2014 (UTC)


 * $(4):$ Use Maple to execute the command below and carefully explain the output.




 * $(5):$ Explain the behavior of the Euler Phi Function in finding all positive solutions to:


 * $(a): \phi(n) = 6, (b): \phi(n) = 14, (c):\phi(n) = 24$.